Stochastic Forcing of the North Atlantic Wind-Driven ... - AMS Journals

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JOURNAL OF PHYSICAL OCEANOGRAPHY

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Stochastic Forcing of the North Atlantic Wind-Driven Ocean Circulation. Part I: A Diagnostic Analysis of the Ocean Response to Stochastic Forcing KETTYAH C. CHHAK

AND

ANDREW M. MOORE

Program in Atmospheric and Oceanic Sciences, and Cooperative Institute for Research in Environmental Sciences, University of Colorado, Boulder, Colorado

RALPH F. MILLIFF Colorado Research Associates, Boulder, Colorado

GRANT BRANSTATOR

AND

WILLIAM R. HOLLAND

National Center for Atmospheric Research,* Boulder, Colorado

MICHAEL FISHER European Centre for Medium-Range Weather Forecasts, Reading, United Kingdom (Manuscript received 20 September 2004, in final form 8 August 2005) ABSTRACT At midlatitudes, the magnitude of stochastic wind stress forcing due to atmospheric weather is comparable to that associated with the seasonal cycle. Stochastic forcing is therefore likely to have a significant influence on the ocean circulation. In this work, the influence of the stochastic component of the wind stress forcing on the large-scale, wind-driven circulation of the North Atlantic Ocean is examined. To this end, a quasigeostrophic model of the North Atlantic was forced with estimates of the stochastic component of wind stress curl obtained from the NCAR Community Climate Model. Analysis reveals that much of the stochastically induced variability in the ocean circulation occurs in the vicinity of the western boundary and some major bathymetric features. Thus, the response is localized even though the stochastic forcing occurs over most of the ocean basin. Using the ideas of generalized stability theory, the stochastically induced response in the ocean circulation can be interpreted as a linear interference of the nonorthogonal eigenmodes of the system. This linear interference process yields transient growth of stochastically induced perturbations. By examining the model pseudospectra, it is seen that the nonnormal nature of the system enhances the transient growth of perturbation enstrophy and therefore elevates and maintains the variance of the stochastically induced circulations in the aforementioned regions. The primary causes of nonnormality in the enstrophy norm are bathymetry and the western boundary current circulation.

1. Introduction The ocean is subject to surface wind stress forcing that drives the ocean general circulation. The wind

* The National Center for Atmospheric Research is sponsored by the National Science Foundation.

Corresponding author address: Dr. Kettyah C. Chhak, Program in Atmospheric and Oceanic Sciences, and Cooperative Institute for Research in Environmental Sciences, University of Colorado, Campus Box 311, Boulder, CO 80309-0311. E-mail: [email protected]

© 2006 American Meteorological Society

JPO2852

stress can be decomposed into two components: a deterministic component (e.g., the seasonal cycle), and a stochastic component (e.g., atmospheric “weather”). At midlatitudes the magnitude of the stochastic component is comparable to that of the seasonal cycle (e.g., Willebrand 1978; Chave et al. 1991; Samelson and Shrayer 1991) and observations indicate that stochastic forcing has a significant influence on the ocean circulation (e.g., Niiler and Koblinsky 1989; Luther et al. 1990; Chave et al. 1991, 1992; Stammer and Wunsch 1999). Because of its potential importance to the ocean circulation, many studies have investigated the effect of atmospheric variability on the ocean. Early works in-

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clude those of Veronis and Stommel (1956), Phillips (1966), and Veronis (1970) who argued that atmospheric variability will produce a primarily barotropic ocean response. Willebrand et al. (1980) used a hierarchy of linear ocean models to investigate the response of the North Atlantic Ocean to observed estimates of large-scale stochastic wind stress forcing. They found the stochastically forced response at periods between 1 and 300 days was primarily barotropic, in agreement with the earlier works. Further studies have also confirmed this result (e.g., Magaard 1977; Leetmaa 1978; Philander 1978; Treguier and Hua 1987; Brink 1989; Samelson 1990; Large et al. 1991; Lippert and Müller 1995; Müller 1997). Other studies have found that small-scale and/or lowfrequency forcing yields a primarily baroclinic response (e.g., Magaard 1977; Lippert and Käse 1985). The influence of stochastic forcing on the temporal and spatial ocean response has been the focus of recent work by Frankignoul et al. (1997), Cessi and Louazel (2001), and Sura and Penland (2002). The role of stochastic forcing in generating ocean eddies has also been the subject of investigation. Using a simplified linear, quasigeostrophic (QG), flat-bottomed, ␤-plane ocean model, and a spectral representation of the stochastic forcing, Frankignoul and Müller (1979) and Müller and Frankignoul (1981) found that local stochastic forcing by the atmosphere reproduces many aspects of the observed eddy field in the North Pacific Ocean and may be an important generating mechanism for eddies in regions of low eddy activity (i.e., the interior ocean). This idea of wind-induced turbulence (i.e., eddy kinetic energy) is supported by numerous other works (e.g., Rhines 1975; Wearn and Baker 1980; Haidvogel and Rhines 1983; Treguier and Hua 1987; Garzoli and Simionato 1990; Fu and Smith 1996; Stammer and Wunsch 1999). While the importance of stochastic wind forcing on the ocean circulation has long been recognized, much of the previous research has been primarily statistical in nature and often the influence of the mean ocean circulation and/or its time dependence on the evolution of stochastically induced variability has been ignored. In many previous studies, the coherence between the stochastic forcing and the ocean response has been examined. However, different phenomena can yield the same spectra and coherence often obscuring the dynamics of the system, particularly in the presence of bathymetry. In this paper we will revisit the problem of stochastic forcing using a new approach, the generalized stability theory (GST). GST circumvents the shortcomings of more traditional approaches to the problem, allowing us to elucidate the dynamics involved. Much of the ground work for GST was laid down by

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Farrell and Ioannou (1996a,b). Traditional stability theory is generally predicated on whether the governing linearized dynamical operators support exponentially growing eigenmodes. If any such eigenmodes exist, then the dynamical system is considered unstable in the limit t → ⬁ (Pedlosky 1987). In dynamical systems governed by nonnormal operators (i.e., systems that possess nonorthogonal eigenmodes), rapid, superexponential transient perturbation growth can occur due to the linear interference of several nonorthogonal eigenmodes, even if all the modes are stable. Nonnormality is often associated with shearing and straining flows in the atmosphere and oceans (Farrell and Ioannou 1993, 1995), and GST provides a complementary geometric view of fluid instability that can be more illuminating than traditional stability analysis alone. This paper is Part I of a two-part study. In this part, our goal is to explore the influence of the stochastic component of the wind forcing on the large-scale winddriven circulation of the North Atlantic Ocean using GST to understand the dynamics involved. The paper is organized as follows. Section 2 describes the ocean model while in section 3 we describe the wind data used in our experiments. A description of the stochastic forcing and the stochastically induced response is explored using a norm equation in section 4. In section 5 the basic theoretical ideas of GST are presented, followed by section 6 where it is shown that the conditions for transient growth are satisfied. A summary of our findings and conclusions are presented in section 7. In the companion paper Chhak et al. (2006, hereinafter referred to as Part II), we extend the GST analysis to examine the perturbations that are excited by the stochastic forcing, which helps to illuminate the stochastically induced variability that arises. The role played by Rossby waves in the stochastically forced system is explored by studying in more detail the features of the linearly interfering nonorthogonal eigenmodes that are excited the most by the stochastic forcing. Part II also discusses the implications of the large number of degrees of freedom the system possesses as well as various sensitivity studies that demonstrate the robust nature of our results.

2. Model description The model used in the present study is the nonlinear QG model of Holland (1978), Schmitz and Holland (1986), and Milliff et al. (1996). The model has five layers each of 1000-m thickness (chosen for mathematical convenience), realistic coastal geometry and bathymetry of the North Atlantic (9°–53°N, 99°–7°W), and a 1° ⫻ 1° horizontal resolution. Our results are

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insensitive to the model level configuration (see Part II, section 5) and the relatively low horizontal resolution was chosen for several reasons. First, we wish to focus on the stochastically induced response in the absence of energetic hydrodynamic instabilities that are common in higher-resolution models. In this way we can study linear eigenmode interference effects in the absence of exponentially growing instabilities. Second, many of the computations described are complex, and may be difficult or computationally prohibitive in much higher resolution models. Third, our primary focus in this study is on linear problems, an important precursor to understanding the nonlinear circulations encountered in higher-resolution models. Following Pedlosky (1996), we write the model equations in their vertically discretized form: ⭸␺n f0 ⭸␨n ⫹ J共␺n, ␨n兲 ⫹ ␤ ⫽ ␦1,n W ⫹ AHⵜ2␨n ⫺ ␦5,nr␨5, ⭸t ⭸x H1 E 共1兲 where n ⫽ 1, . . . , 5 refers to each vertical level of thickness Hn; ␺n is the streamfunction; ␨n ⫽ ⵜ2␺n ⫺ f 20/Hn[(␺n ⫺ ␺n⫺1)/g⬘n⫺1 ⫺ (␺n⫹1 ⫺ ␺n)/g⬘n] is the vorticity for n ⫽ 2, 3, 4 while ␨1 ⫽ ⵜ2␺1 ⫺ f 20(␺1 ⫺ ␺2)/ (g⬘1H1), and ␨5 ⫽ ⵜ2␺5 ⫺ f 20(␺5 ⫺ ␺4)/(g⬘4H5) ⫹ f0Zb/H5; the Jacobian operator is defined by J(␺n, ␨n) ⫽ ⳵␺n/ ⳵x⳵␨n/⳵y ⫺ ⳵␺n/⳵y⳵␨n/⳵x; WE is the surface Ekmanpumping velocity due to wind stress curl and is nonzero only in the surface layer; g⬘n ⫽ (␳n⫹1 ⫺ ␳n)g/␳0 are the reduced gravities of each layer of density ␳n; f0 ⫽ 2⍀ sin(⌰0), ⍀ ⫽ 7.292 ⫻ 10⫺5 rad s⫺1 is the angular velocity of the earth; ⌰0 ⫽ 31° is the central latitude of the model domain; Zb(x, y) describes the bathymetry; ␦i,n is the Kronecker delta function; r ⫽ 1 ⫻ 10⫺7 s⫺1 is the coefficient of bottom friction; and AH ⫽ 3200 m2 s⫺1 is the horizontal diffusion coefficient. The boundary conditions are free slip (␨n ⫽ 0) and no normal flow at the boundary, meaning ⳵␺/⳵s ⫽ 0 along the boundary where s is the tangential direction to the boundary. Following Milliff and McWilliams (1994), an integral consistency constraint is also imposed along the solid boundaries in the model to ensure QG mass balance (McWilliams 1977; Pinardi and Milliff 1989; Milliff and McWilliams 1994). Equation (1) is solved by decomposing ␺ into vertical normal modes, denoted ␾m with corresponding eigenvalues ␭m. The solution for each vertical mode consists of two parts so that ␾m ⫽ ␾ˆ m ⫹ cm(t)␾m * , where ␾ˆ m ⫽ 0 along the boundary, and ␾m * is the solution of (ⵜ2 ⫺ ␭m)␾* * ⫽ m ⫽ 0 such that ␾m 1 along the boundary. The time-dependent constant cm(t) is governed by ⳵cm(t)/⳵t ⫽ ⫺兰兰O ⳵␾ˆ m/⳵t dx dy/兰兰O ␾m * dx dy , where 兰兰O dx dy denotes an integral over the

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entire ocean domain. From the continuity equation, the vertical velocity wn ⫽ ⳵zn/⳵t ⫹ J(␺n, zn), where zn ⫽ f0(␺n ⫺ ␺n⫺1)/g⬘n⫺1 is the z coordinate of the top of level n. At the ocean surface w ⫽ WE and at the bottom w ⫽ J(␺5, Zb). The top and bottom boundary conditions are implicit in the definitions of ␨1 and ␨5 above. The chosen value of AH yields a mean circulation with a realistic shape but poorly resolves the western boundary layer of thickness (AH/␤)1/3. The sensitivity of the model results to variations in AH is discussed in section 4b, where it is shown that the basic conclusions of this work are not influenced by the choice of AH. Therefore, we present results for the lowest value considered and associated with the most realistic circulation. The stratification described by g⬘n was chosen to be that typical of the North Atlantic as described by Schmitz and Holland (1986). Full-amplitude bathymetry of the North Atlantic was used rather than rescaling the amplitude as often done in QG models (e.g., Milliff and Robinson 1992). Comparisons of primitive equation and QG models (Semtner and Holland 1978; Spall 1988; Spall and Robinson 1989) using full-amplitude bathymetry reveal that both produce remarkably similar circulation patterns despite the fact that the QG assumption of small-amplitude bathymetry is not satisfied.

3. Ocean wind forcing The focus of this work is the impact of stochastic wind forcing on the ocean circulation. Our choice of wind forcing was therefore motivated by the need to decompose the ocean Ekman-pumping velocity WE into deterministic and stochastic components, a task complicated by interannual variations in surface boundary conditions. To minimize these effects, daily surface wind data were obtained from a 100-yr integration of the National Center for Atmospheric Research (NCAR) Community Climate Model, version 3, (CCM3) in which the same annual cycle of SST was imposed each year. The CCM3 employs a horizontal T42 spectral resolution with a transform grid that is approximately 2.8° ⫻ 2.8°, and there are 18 vertical levels that use hybrid coordinates (Kiehl et al. 1996). The 10-m zonal and meridional wind components were interpolated onto the ocean model grid. The efficacy of the CCM3 winds was checked by comparing the 100-yr climatological mean surface wind fields with the observed long-term means of the same fields from the Comprehensive Ocean Atmosphere Data Set (COADS; not shown). Overall, the CCM3 surface wind fields are a reasonable representation of the observed surface winds over the North Atlantic.

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FIG. 1. Variance of the stochastic component of Ekman-pumping velocity wE as a function of lat and lon. The contour interval used is 3.0 ⫻ 10⫺11 m2 s⫺2 and contour labels are scaled by 10⫺11 m2 s⫺2.

The winds were converted to surface wind stress using the drag law of Milliff and Morzel (2001) and the speed-dependent drag coefficient of Large et al. (1994). The associated Ekman-pumping velocity, WE was decomposed into a deterministic component, denoted by wE, and a stochastic component, denoted by wE. There is no unique way to perform this decomposition, and numerous approaches are reported in the literature (e.g., Frankignoul and Müller 1979; Willebrand et al. 1980; Rienecker and Ehret 1988). We have followed the approach of Kleeman and Moore (1997) and define the stochastic forcing wE as that component of Ekmanpumping velocity that is not associated with variations in SST, and due only to internal variability of the atmosphere. As noted above, the main motivation for using the CCM3 dataset is that there are no interannual variations in SST, unlike other available wind datasets such as National Centers for Environmental Prediction (NCEP)–NCAR or the European Centre for MediumRange Weather Forecasts (ECMWF) reanalysis products. Therefore, removal of the seasonal cycle from the CCM3 winds yields the desired stochastic component. To this end, a daily climatology of Ekman-pumping velocity was first removed to yield Ekman-pumping velocity anomalies. In addition, the first three harmonics of the anomalies were also removed following the usual practice (Wilks 1995). The resulting anomalies were then assumed to compose only the Ekman-pumping velocity due to the internal variability of the atmosphere that is not associated with simultaneous variations in

SST, our working definition of wE. The deterministic component of the wind wE is then given by WE ⫺ wE. Note that in Part II of this paper, we will also discuss how wE can be separated into a spatial- and timevarying component, where the time-varying component represents a lag-1 autoregressive model. This kind of separation facilitates the use of GST to examine the response of the ocean circulation to stochastic forcing as detailed in Part II. The average decorrelation time of the resulting stochastic component of Ekman-pumping velocity, wE, was found to be ⬃2 days, which is consistent with the time scales of internal atmospheric variability. Figure 1 shows the variance of wE as a function of latitude and longitude, and indicates that the stochastic forcing variance increases with latitude and coincides with the North Atlantic storm track. At 42°N, the longitudinally averaged standard deviation of wE is 9.8 ⫻ 10⫺6 m s⫺1 as compared with a value of 1.3 ⫻ 10⫺6 m s⫺1 inferred from wE near the center of the North Atlantic subtropical gyre, indicating that wE dominates WE at midlatitudes. This agrees qualitatively with observations (e.g., Samelson and Shrayer 1991).

4. Stochastically induced variability The focus of the present study is the linear response of the ocean circulation to stochastic forcing. To this end, consider (1) subject to wE ⫽ WE ⫺ wE, where the resulting circulation is denoted ␺B n . Consider now, (1) subject to WE ⫽ wE ⫹ wE, where we denote by ␦␺n the

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FIG. 2. (a) The annual mean basic-state streamfunction ␺B 1 where shaded and unshaded regions are of opposite sign. The contour interval is 3 ⫻ 103 m2 s⫺1. (b) Contours of the bottom depth (km) where the contour interval is 0.4 km.

stochastically induced perturbations to ␺B n resulting from wE. The time evolution of these perturbations during the early stages of development when perturbations are linear is described by ⭸␦␺n f0 ⭸␦␨n ⫽ ⫺J共␺nB, ␦␨n兲 ⫺ J共␦␺n, ␨nB兲 ⫺ ␤ ⫹ ␦1,n w ⭸t ⭸x H1 E ⫺ ␦5,nr␦␨5 ⫹ AHⵜ2␦␨n,

共2兲

with associated boundary conditions and consistency constraint. Equation (2) is referred to as the tangent linear model (TLM) and is based on the first-order linearization of (1) about ␺B n . Since the QG model solves a discretized form of (1), we will denote (2) symbolically as d␦␨Ⲑdt ⫽ A␦␨ ⫹ f0wEⲐH1,

共3兲

where ␦␨ denotes the vector of model perturbation vorticity gridpoint values, wE is the vector of stochastic forcing and is nonzero only in level 1, and A is the discretized tangent linear operator in (2). Equation (2) was solved using the same procedures used to solve (1). The primary focus of the present study is systems for which A is asymptotically stable, in which case solutions of (3) will reach a statistical equilibrium. To ensure stable circulations, it was necessary to scale wE by a factor of 0.71. This scaling was used in all of the experiments reported here. An alternative approach would be to use a scaling of 1 for wE and increase r and/or AH in (1), but this was not done here. The annual mean of ␺B 1 resulting from the rescaled wE is shown in Fig. 2a and takes the form of a welldeveloped subtropical gyre. However, because of the

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FIG. 3. The dashed line is ⌬Q from the stochastically forced nonlinear model. The solid line is Q from the stochastically forced TLM linearized about the annual mean basic state shown in Fig. 2a. The time mean of Q, denoted Q ⫽ 1.03 ⫻ 105 m3 s⫺2, is also indicated.

relatively coarse horizontal resolution of the model, the maximum Gulf stream velocity is only ⬃0.1 m s⫺1, which is very weak relative to the observed maximum ⬃2 m s⫺1. The weak subpolar gyre is associated with the location of the northern boundary, although a more realistic circulation can be obtained by enhancing the Ekman-pumping velocity in this region as described by Milliff et al. (1996; see also Part II, section 5a). For later reference, the model bathymetry is shown in Fig. 2b. The circulation, ␺B n is time dependent and renders A in (3) nonautonomous. However, after a few years, ␺B n settles into a seasonal cycle that repeats each year, and experiment reveals little difference between the stochastically induced variability of (3) when A is autonomous and nonautonomous. Therefore, to simplify the analyses presented in subsequent sections, the annual mean of ␺B n was used to solve (3).

a. Validity of the tangent linear assumption Before proceeding further, it is necessary to demonstrate the validity of (2). Because of the relatively coarse horizontal resolution used in the present study, the energetics of the system are relatively uninteresting because of the sluggish circulation. For a number of reasons we will instead consider the perturbation enstrophy norm as a measure of the stochastically induced variance. First, because of the QG nature of the model, vorticity dynamics play an important role in controlling the circulation and clearly enstrophy is an indicator of the physical processes at work. Second, as we will dem-

onstrate in section 5, sources and sinks of enstrophy can be identified with the nonnormal aspects of the governing operator A relative to the perturbation enstrophy norm. The sources of enstrophy are essentially Rossby wave generators so the enstrophy norm provides a clear picture of the role played by Rossby wave dynamics in controlling stochastically induced variability. There are no corresponding Rossby wave sources when a perturbation energy norm is used. Third, enstrophy plays a critical role in the turbulent cascade, so understanding its properties and behavior in the large-scale, linear regime, as we do here, is an essential first step to understanding the behavior of more complex, nonlinear regimes encountered in eddy-resolving models, the subject of a future study. Energy is also important in this regard, but is less interesting in the linear regime considered here. The energetics of stochastically induced variability is of greater interest in eddy-resolving models and is also the subject of our future study. The validity of the tangent linear assumption was tested as follows. Two 100-yr runs of the nonlinear QG model described by (1) were performed: in one run the model was forced with WE ⬅ wE, and in the other the model was forced with WE ⬅ wE ⫹ wE. A time series of the basin-integrated perturbation enstrophy ⌬Q ⫽ 5 2 1⁄2兰兰 O 兺n⫽1Hn(⌬␨n) dx dy computed from the difference ⌬␨n between the two model solutions is shown in Fig. 3, where as before 兰兰O dx dy denotes an area integral over the North Atlantic Ocean basin. In a third experiment, the TLM was run for 100 yr linearized about ␺B n and forced by wE. A time series of the per-

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FIG. 4. A power spectrum of Q in Fig. 3 computed using a maximum entropy method of order 2.

turbation enstrophy Q ⫽ 1⁄2兰兰O 兺5n⫽1Hn(␦␨n)2 dx dy is also shown in Fig. 3. Figure 3 indicates that ⌬Q and Q track each other very well and demonstrates that the tangent linear assumption is justified.

b. A diagnostic analysis of stochastically induced variability Figure 3 reveals that after an initial spinup period of about 20 yr, Q reaches a statistically steady average value (i.e., the time series is stationary). A power spectrum of Q computed using the maximum entropy method (MEM) of the order 2 (Dobrovolski 2000) is shown in Fig. 4 and the spectrum is relatively insensitive to the order of the MEM chosen (we explored up to the order of 50). For periods ⬍1 yr the spectral slope is ⬃⫺4. This cutoff period is considerably longer than that exhibited by shallow-water and primitive equation models and observations (e.g., Willebrand et al. 1980) probably because QG dynamics preclude higherfrequency responses associated with Kelvin waves and inertia–gravity waves. The spectral slope, however, is consistent with those of Willebrand et al. (1980), who found spectral slopes between ⫺2 and ⫺4 but at considerably shorter periods. There are no obvious resonance peaks in Fig. 4, which is most likely due to the fact that the dissipation time of individual modes is very much less than the basin transit time of Rossby waves, meaning that resonant basin modes cannot be established (Willebrand et al. 1980). In addition, bathymetry may introduce very localized topographic resonances that are masked by the basin-integrated enstrophy norm used here. The time evolution of Q in Fig. 3 is described by

兺冕冕 5

dQ ⫽⫺ dt n⫽1

Hn␦␨nun · ⵱共ⵜ2␺nB兲 dx dy

O

兺冕冕

Qrelative

5

⫹

n⫽1

⫺

␦␨nf 20un · ⵱␥nB dx dy

O

冕冕

Qstretch

␦␨5 f0u5 · ⵱Zb dx dy ⫹

O

冕冕

␦␨1 f0wE dx dy

O

Qbathy

兺 H 冕冕

Qforce

5

⫺

n

n⫽1

AH⵱␦␨n · ⵱␦␨n ⫹ ␦5,nr␦␨n2 dx dy

O

Q diss

兺 H ␤ 冖兵共␷ ⫺ u 兲Ⲑ2 dy ⫹ u ␷ dx其 , 5

⫺

2 n

n

2 n

n n

n⫽1

C

Qbndy

共4兲

B where un ⫽ ⫺⳵␦␺n/⳵y and ␷n ⫽ ⳵␦␺n/⳵x; ␥B n ⫽ (␺n ⫺ B B B ␺n⫺1)/g⬘n⫺1 ⫺ (␺n⫹1 ⫺ ␺n )/g⬘n for n ⫽ 2, 3, 4, while ␥B 1 ⫽ B B B B ⫺ ␺ )/g⬘ H , and ␥ ⫽ ( ␺ ⫺ ␺ )/g⬘ H ; 养 indicates (␺B 1 2 1 1 5 5 4 4 5 C a counterclockwise line integral around the ocean boundary C. Each term in (4) represents a physical process that serves as a source and/or sink of enstrophy that contributes to the time rate of change of Q. The first term in (4), denoted Qrelative, represents enstrophy sources/

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sinks proportional to the perturbation momentum stresses associated with the straining component of ␺B n (Rhines 1977). Here Qrelative will be largest in regions of the ocean where the basic-state circulation has large relative vorticity gradients such as near the western boundary current. The second term, Qstretch, represents sources/sinks of enstrophy due to perturbations that are stretched or compressed as they move relative to the deformed pressure surfaces described by ␺B n . There are three special cases for which Qstretch will vanish. If the basic-state circulation is barotropic, then ␥B n ⫽ 0 for all n, and Qstretch vanishes since there is no relative deformation of the basic-state pressure surfaces relative to each other. If the perturbation flow ␦␨, is barotropic then ␦␨nun is independent of depth in which case Qstretch ⫽ 0 since 兰兰O 兺5n⫽1␥B n dx dy ⫽ 0 , which represents no net stretching or compression of the basic-state fluid columns. Last, if the ␥B n are homogeneous in the x and y, then Qstretch ⫽ 0 by virtue of the no-normal-flow boundary condition. The third term, Qbathy, represents the enstrophy sources due to the stretching of perturbations as they move over the bathymetry. This term will include enstrophy sources due to topographic Rossby waves. The physical interpretation of this term is similar to that of Qstretch except the stretching and compression of perturbations is relative to the bathymetry. The fourth term, Qforce, represents the sources/ sinks of enstrophy due to the stochastic forcing. The fifth term, Qdiss, represents the sinks of enstrophy due to horizontal eddy viscosity and bottom friction. The last term, Qbndy, represents the net sources and sinks of enstrophy at continental boundaries due to Rossby wave reflections. Short (long) Rossby waves emanating from a western (eastern) boundary represent a source (sink) of enstrophy. Time series of the time rate of change of Q associated with each term in (4) from the stochastically forced TLM are shown in Fig. 5. Figure 5 indicates that Qrelative and Qbndy are negligible when compared with the other terms and so they are not considered further. Figure 5 also shows that Qbathy contributes most to dQ/dt. Sinks of enstrophy due to dissipation typically occur on all space scales over the entire ocean basin and are largest near the coast, especially near the western boundary. The source of enstrophy from the stochastic forcing occurs over much of the northern half of the basin and mirrors the forcing variance shown in Fig. 1 in that it increases with latitude and is largest in the open ocean. Even though the forcing occurs over a broad range of longitudes of the ocean basin, Qbathy ⫹ Qstretch is mainly associated with stochastically induced variability that is confined to the western boundary and the Gulf Stream as well as parts of the Caribbean Sea and

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Gulf of Mexico as shown in Fig. 6a, which shows the root-mean-square of the integrand of these terms in (4). This localized response occurs for two reasons: first, there is a decrease in ocean depth along the western boundary as shown in Fig. 2b, so stochastically induced perturbations moving over the shelf will induce variance as the water column acquires and loses vorticity (cf. Fig. 5f); and second, the deformation of the basicstate circulation pressure surfaces along the western boundary is also a contributing factor as indicated by Qstretch in Fig. 5e. The tendency for stochastically induced variance to accumulate along the western boundary has been noted in other studies (e.g., Willebrand et al. 1980; Lippert and Käse 1985). From analysis of the time series of each term in (4), it is apparent that the bathymetry and the basic-state flow ␺B n are responsible for a substantial fraction of Q. Figure 6a reveals that the western boundary region accounts for much of the perturbation enstrophy variance. Therefore, although the stochastic wind forcing occurs at all longitudes primarily at mid- and high latitudes, a significant fraction of the resulting variance in the circulation occurs in the vicinity of the western boundary. The dynamics of this response will be explored further in section 5 and in Part II. As discussed in section 2, the value of AH ⫽ 3200 m2 ⫺1 s used throughout this paper poorly resolves the western boundary layer in the model. To explore the sensitivity of the model results to this issue, we repeated our calculations using values of AH ⫽ 3 ⫻ 104 and 2.2 ⫻ 105 m2 s⫺1. In the former case, the western boundary layer thickness ␦m ⫽ (AH/␤)1/3 ⫽ ⌬x, and in the latter case, ␦m ⫽ 2.1⌬x, where ⌬x is the model grid spacing. Figures 6b,c show the rms of Qbathy ⫹ Qstretch ⫹ Qrelative from the stochastically forced TLM in each case, respectively. As expected, the increase in ␦m widens the region of stochastically induced variance along the western boundary. Interestingly, we note that other regions are favored and associated with the Mid-Atlantic Ridge and other bathymetric features. These regions are also favored in the case where AH ⫽ 3200 m2 s⫺1, although they are less prominent.

5. Linear interference of eigenmodes In section 4, we examined the stochastically induced response of the system using the norm evolution of Eq. (4). In this section, we will explore the dynamics of the stochastically induced response in more detail using the ideas of GST to interpret the sources and sinks in (4) in terms of linear interference of the eigenmodes of the TLM (3). As we shall see, this is a very enlightening geometrical interpretation of stochastically induced

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FIG. 5. The time series of each term identified in (4) from the stochastically forced TLM: (a) Qforce, (b) Qdiss, (c) Qbndy, (d) Qrelative, (e) Qstretch, and (f) Qbathy. The rms values of each time series are also indicated.

variance. We present here only a brief review of the important ideas. Those readers unfamiliar with GST should consult Farrell and Ioannou (1993, 1996a,b) and Kleeman and Moore (1997) for more details. The stochastically induced response of a nonnormal, linear system to stochastic forcing is of interest because

the stochastic forcing acts as a source of continuous perturbations that can undergo transient growth due to linear eigenmode interference. This transient growth can lead to an increase in the stochastically induced variability. To illustrate this idea, we will denote the ␦␨k, so that Aˆ ␦␨k ⫽ ␴kˆ ␦␨k is eigenvectors of A in (3) as ˆ

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strophy is Q ⫽ 1⁄2(␦␨HX␦␨) ⬅ 1⁄2兺5n⫽1Hn兰兰O ␦␨2n dx dy, where superscript H denotes the conjugate transpose, and X is a diagonal matrix with elements given by the model grid cell volumes. Expanding in terms of the eigenmodes of A, Eq. (4) can be written as dQ ⫽ dt

N

N

兺兺 ℜe关a*共t兲a 共t兲␴ ˆ␦␨ i

j

j

H ˆ i X␦␨j兴

i⫽j j⫽i

Qbathy ⫹ Qstretch ⫹ Qrelative ⫹ Qbndy N

⫹

兺 |a 共t兲|

2

i

ℜe共␴i兲 ⫹ ␦␨HXf共t兲

i⫽1

Qdiss

Qforce,

共5兲

where we have assumed that the eigenmodes are norˆ ␦␨H malized so that ˆ k X␦␨k ⫽ 1. As indicated in (5) the sum of several terms in (4) can be represented in terms ˆ ␦␨H of the linear superposition (i.e., ˆ i X␦␨j) of the eigenmodes. Recall that we are considering asymptotically stable systems so ℜe(␴i) ⬍ 0 for all i, hence Qdiss ⬍ 0, as required (ℜe indicates real). For a normal system ˆ ˆ ␦␨H i X␦␨j ⫽ ␦ij and Qbathy ⫹ Qstretch ⫹ Qrelative ⫹ Qbndy ⫽ 0, so the stochastically induced variance is given by the balance between Qforce and Qdiss. For a nonnormal system, the eigenmodes are not orthogonal, so in general ˆ ˆ ␦␨H i X␦␨j ⫽ 0 for all i ⫽ j. Hence in this case, Qbathy ⫹ Qstretch ⫹ Qrelative ⫹ Qbndy ⫽ 0 and additional sources and sinks of stochastically induced variance exist. It is these ideas that form the basis of GST. A simple illustrative example of these ideas is presented in the appendix.

6. Demonstration of transient growth

FIG. 6. The rms of the time-averaged integrand of (a) Qbathy ⫹ Qstretch, and Qbathy ⫹ Qstretch ⫹ Qrelative in (4) for (b) AH ⫽ 3 ⫻ 104 m2 s⫺1 and (c) AH ⫽ 2.2 ⫻ 105 m2 s⫺1 as a function of latitude and longitude from the stochastically forced TLM. The contour interval used in (a) is 0.9 ⫻ 10⫺18 s⫺3 and contour labels are scaled by 1 ⫻ 10⫺18 s⫺3. In (b) and (c) the contour labels are scaled by 1 ⫻ 10⫺19 s⫺3and contour intervals are 1.5 ⫻ 10⫺19 s⫺3 and 0.5 ⫻ ⫺19 10 s⫺3, respectively.

an eigenvalue equation, where ␴k are the associated eigenvalues. Expanding in terms of these eigenvectors, ˆ we have ␦␨(t) ⫽ 兺N i⫽1ai(t)␦␨i, where ai(t) are time-dependent amplitude coefficients. The perturbation en-

An important idea central to this paper and its companion (Part II) is that transient growth of stochastically induced perturbations is responsible for maintaining the variance shown in Fig. 3 via the linear eigenmode interference described by (5). In this section we will demonstrate that the necessary conditions exist for transient growth to occur. To this end, we will explore the stochastically induced response of the system as a function of forcing frequency. Following Trefethen et al. (1993) consider (3) driven by f0wE/H1 ⫽ he␻t, where ␻ is complex and h is an arbitrary vector. Since A is asymptotically stable, then as t → ⬁, ␦␨ ⫽ he␻t/(␻I ⫺ A). In this limit, the ratio of the size of the response to the size of the forcing at any frequency ␻ is of interest and given by ||␦␨||/||he␻t|| ⱕ ||K(␻)||, where K(␻) ⫽ (␻I ⫺ A)⫺1 is the resolvent matrix of (3). Following Trefethen et al. (1993), it can be shown that 1/dist[␻, ⌳(A)] ⱕ ||K(␻)|| ⱕ ␬(E)/dist[␻, ⌳(A)], where dist[␻, ⌳(A)] represents the shortest dis-

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tance in the complex plane between ␻ and the spectrum of eigenvalues of A denoted ⌳(A), and ␬(E) is the condition number of the matrix of eigenvectors E of A and describes the nonorthogonality of the eigenmodes of A; the more nonorthogonal (or nonnormal) the eigenmodes, the larger ␬(E) will be. In a normal system, the columns of E are orthogonal and ||K(␻)|| ⫽ 1/dist[␻, ⌳(A)]. In this case, the system resonates only when ␻ is close to the spectrum ⌳(A) (Trefethen 1997). For a nonnormal system ␬(E) ⱖ 1 because some or all of the columns of E are nonorthogonal, with the result that the amplitude of the response can be large for forcing frequencies distant from ⌳(A) (Trefethen et al. 1993). Contours of ||K(␻)|| in the complex ␻ plane are called pseudospectra. As the degree of nonormality of the eigenmodes increases so then does ␬(E), which in turn leads to an increase in the upper bound for ||K(␻)|| according to the above inequality. Figure 7a shows ||K(␻)|| versus ␻ for the physically relevant case of ℜe(␻) ⫽ 0 for the Q norm following Wright and Trefethen (2001). For comparison, 1/dist[␻, ⌳(A)] versus ␻ is also plotted, which corresponds to the response of a normal system with the same eigenspectrum ⌳(A). Figure 7a reveals that ||K(␻)|| differs significantly from 1/dist(␻, ⌳(A)), and confirms that the system is nonnormal. This means that the eigenmodes of the system are nonorthogonal, and therefore transient growth of perturbations is due to the linear interference of these nonnormal eigenmodes. Note that the largest response in Fig. 7a occurs at low frequencies, in agreement with Fig. 4. In the absence of forcing, solutions of the TLM (3) can be written in compact form as ␦␨(t) ⫽ R(0, t)␦␨(0), where R(0, t) is the propagator of the system. For an asymptotically stable system, the consistency properties of norms requires that transient perturbation growth is only possible [i.e., ||␦␨(t)||/||␦␨(0)|| ⬎ 1] if ||R(0, t)|| ⬎ 1. Since R(0, t) forms a contraction semigroup, it is subject to the Hille–Yosida theorem, which states that for real positive frequencies [ℜe(␻) ⬎ 0], ||R(0, t)|| ⱕ 1 only if ␧ ⫽ ||K(␻)||⫺1 ⱖ ℜe(␻) (Goldstein 1985). Conversely, for ||R(0, t)|| ⬎ 1 and transient growth, we require ␧ ⬍ ℜe(␻). If we denote by ⌳␧(A) a contour of constant ␧ in the complex ␻ plane, then there will exist a set of frequencies ␻ that all lie on this contour. Considering the extremum of this set shows that transient growth is possible if sup␻∈⌳␧(A)ℜe(␻) ⬎ ␧, where sup␻∈⌳␧ℜe(␻) denotes the largest value of ℜe(␻) along such a contour, where as noted above ℜe(␻) ⬎ 0. Figure 7b shows sup␻∈⌳␧(A)ℜe(␻) versus ␧ for the present case and indicates that the condition for nonnormal transient growth of Q is satisfied. For the enstrophy norm, ||R(0, t)|| corresponds to the

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FIG. 7. (a) The norm of the resolvent ||K(␻)|| vs Im(␻) when ℜe(␻) ⫽ 0 (solid line) and l/dist(␻, ⌳(A)) vs Im(␻) (dashed line), (b) sup␻∈⌳␧(A) ℜe(␻) vs ␧ ⫽ ||K(␻)||⫺1 (solid line) and the 1:1 ratio (dashed line) [above the dashed line ||R(0, t)|| ⬎ 1, while below the dashed line ||R(0, t)|| ⱕ 1], and (c) ||R(0, t)|| vs t for an ocean with the basic state indicated in Fig. 2a (solid line), a basic state of rest (dashed line), and a basic state of rest and uniform ocean depth (dashed–dotted line).

largest singular value of R(0, t). The possibility for nonnormal transient growth was verified by computing the singular values of the propagator R of (3). In the linear limit, the corresponding singular vectors are the fastest growing of all possible perturbations with respect to the enstrophy norm (Farrell and Ioannou 1996a). Figure 7c shows ||R(0, t)|| versus t for the basic state of Fig. 2a, and indicates that transient growth [||R(0, t)|| ⬎ 1] is indeed possible over a wide range of t, even though A is asymptotically stable. Figure 7c also shows ||R(0, t)|| versus t for an ocean

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311

FIG. 8. The initial and final vorticity patterns for layer 1 (n ⫽ 1) associated with the fastest-growing singular vector of the enstrophy norm for a resting ocean of uniform depth at (a) t ⫽ 0 and (b) t ⫽ 5 days, and for a resting ocean with bathymetry at (c) t ⫽ 0 and (d) t ⫽ 50 days. Shaded and unshaded regions are of opposite sign, the contour interval is arbitrary, and the perturbation amplitude growth factor, ||R(0, t)||, for each pattern is indicated.

with a basic state of rest with bathymetry and for a resting ocean of uniform depth; the difference between the three curves illustrates that the sources of nonnormality that make transient growth of Q possible are gradients in the bathymetry, regions of potential vorticity gradients associated with circulation features like the Gulf Stream, and continental boundaries, all of which act as Rossby wave generators. Figure 7c indicates that the bathymetry is the dominant source of nonnormality in the present case. Figures 8a,b and 8c,d show, respectively, the initial and final time structures of the fastest growing singular vector over the indicated optimal growth times for a resting ocean of uniform depth and a resting ocean with bathymetry. Figures 8a,b illustrate how the western boundary plays a role in promoting enstrophy growth via Qbndy by the reflection of long Rossby waves to generate short Rossby waves, although Fig. 5 shows

that this process is not important in the stochastically forced case considered here. Figures 8c,d illustrate how the bathymetry creates enstrophy by exciting topographic Rossby waves with wavelengths shorter than the initial perturbations. Following (5), the singular vector growth in Fig. 8 can be interpreted as due to the linear interference of the nonorthogonal eigenmodes of the system. We will demonstrate in Part II that some of these eigenmodes are in the form of Rossby wave packets that account for the enstrophy variance shown in Fig. 6. The singular vectors of the full system in the 30–55-day optimal growth time range are very similar to the optimal forcing patterns examined in Part II.

7. Summary and discussion In this work, we have explored the response of the QG wind-driven ocean circulation in the North Atlantic

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to stochastic variations in surface wind stress forcing. In contrast to previous studies, the ideas of GST were used here to study the response of the system from a geometric perspective rather than using the more traditional approach where the coherence between the forcing and the response is most often considered. Specifically, we found that the conditions for nonnormal transient growth demanded by GST are satisfied, the vehicle for such growth being linear interference of the nonorthogonal eigenmodes of the underlying circulation. While the stochastic forcing is generally basinwide, the stochastically induced response is more localized and favors the western boundary and some topographic features such as the Mid-Atlantic Ridge, in agreement with other studies (e.g., Willebrand et al. 1980; Lippert and Käse 1985; Cessi and Louazel 2001). As noted, many studies have also focused on the coherence between stochastic forcing and the ocean response, although bathymetry tends to destroy or confuse any such coherences thereby hampering the dynamical interpretation of the observed response. The present study illustrates how the intrinsic nonnormality of the underlying flow conspires to create these effects, and identifies the role played by gradients in bathymetry and gradients in potential vorticity of the western boundary current in rendering the system nonnormal. The idea that bathymetry plays a role in wave amplification is supported by observations in the North Atlantic. For example, Osychny and Cornillon (2004) used satellite observations of sea surface height to detect westward-propagating waves and note that the largest amplitudes are found in the vicinity of the Gulf Stream particularly where the stream crosses the New England Seamounts and the Newfoundland Ridge, two bathymetric features near the western boundary. They also note that wave fronts do not cross the Mid-Atlantic Ridge from the eastern boundary so large amplitude waves observed west of the ridge must originate and amplify near the western boundary. The stochastically induced variance illustrated in Fig. 6 is in qualitative agreement with these observations. Earlier work by Willebrand et al. (1980) also highlights the potential importance of bathymetry in controlling the stochastically induced response of the ocean by destroying the coherence between the forcing and the response. They, however, concentrated mainly on the large-scale barotropic response of the ocean and found that the stochastically induced response was composed mainly of topographic Rossby waves. In Part II of this study, we explore in detail the dynamical response of the stochastically forced system using further ideas of GST. Specifically, we examine the

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perturbations that are naturally excited by the stochastic forcing and use these perturbations to understand the stochastically induced variability that arises. We also determine the qualitative and quantitative features of the linearly interfering nonorthogonal eigenmodes that are excited the most by the stochastic forcing, which in turn allows us to determine the specific role played by Rossby waves in the stochastically forced system. The results of a number of sensitivity studies are also presented, which demonstrate the robust nature of our results. Acknowledgments. The research described here was supported by a grant from the National Science Foundation Physical Oceanography Program (OCE0002370). Any opinions, findings, and conclusions or recommendations expressed here are those of the authors and do not necessarily reflect the views of the National Science Foundation. We are also grateful to Bob Miller for enlightening discussions on some of the material presented here.

APPENDIX An Illustrative Example of Transient Growth Associated with the Interference of Nonorthogonal Eigenmodes The ideas underpinning transient growth of perturbations due to linear interference of nonorthogonal eigenmodes described in section 5 are best demonstrated by a simple illustrative example (see also Farrell and Ioannou 1996a). Consider the two-dimensional system ds/dt ⫽ As in which A⫽

冋

␴1

共␴2 ⫺ ␴1兲cot␦

0

␴2

册

,

where ␴1 and ␴2 are the eigenvalues of the matrix A, and ␦ is the angle subtended by the corresponding eigenvectors. For a given time interval ␶, the fastest growing of all possible perturbations are the eigenvectors of RT(0, ␶)R(0, ␶), where R(0, ␶) ⫽ eA␶ is the propagator, and perturbation growth is measured in terms of the L2 norm sT(␶)s(␶). Figure A1 shows the growth factor ␮ ⫽ sT(␶)s(␶)/sT(0)s(0) versus ␦ of the fastest growing perturbation, which coincides with the dominant singular vector, for the case of ␶ ⫽ 4, ␴1 ⫽ ⫺0.05, and ␴2 ⫽ 10␴1. In this example, both eigenmodes are asymptotically stable, and have disparate decay rates. Figure A1 shows that when the system is normal (␦ ⫽ ␲/2) no perturbation growth is possible since ␮ ⬍ 1. As ␦ → 0 or ␲, Fig. A1 indicates that ␮ → ⬁ showing that very large transient growth of perturbations is possible

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FIG. A1. The growth factor ␮ of the L2 norm of s vs the angle subtended by the corresponding eigenmodes ␦ for the fastest growing singular vector of the 2 ⫻ 2 illustrative example described in the appendix for ␴1 ⫽ ⫺0.05, ␴2 ⫽ 10␴1, and ␶ ⫽ 4.

when the two eigenmodes of A are very close to parallel or antiparallel. The case of ␦ → 0 or ␲ indicates a situation in which the two eigenmodes are highly nonorthogonal to each other, and the transient growth is the direct consequence of their mutual interference as illustrated in Fig. A2, which shows the time evolution of the singular vector when ␦ ⫽ 4␲/5. It can also be shown that the interference of two complex nonorthogonal eigenmodes with similar (disparate) dissipation rates but disparate (similar) oscillation frequencies can also yield rapid transient growth. In this case, the minimum system dimension required is four since the eigenmodes of A occur in complex conjugate pairs in the physically relevant case when A is real. REFERENCES

FIG. A2. A schematic representation of the evolution of the fastest-growing singular vector depicted in Fig. Al when ␦ ⫽ 4␲/5. Snapshots of the singular vector and its projection on the eigenvectors s1 and s2 are shown for ␶ ⫽ (a) 0, (b) 2, (c) 4, and (d) 6. The value of the L2 norm denoted E is also indicated for each time.

Brink, K., 1989: Evidence for wind-driven current fluctuations in the western North Atlantic. J. Geophys. Res., 94, 2029–2044. Cessi, P., and S. Louazel, 2001: Decadal oceanic response to stochastic wind forcing. J. Phys. Oceanogr., 31, 3020–3029. Chave, A. S., D. S. Luther, and J. H. Filloux, 1991: Variability of the wind stress curl over the North Pacific: Implications for the oceanic response. J. Geophys. Res., 96, 18 361–18 379. ——, ——, and ——, 1992: The barotropic electromagnetic and pressure experiment: 1. Barotropic current response to oceanic forcing. J. Geophys. Res., 97, 9565–9593. Chhak, K., A. M. Moore, R. F. Milliff, G. Branstator, W. R. Holland, and M. Fisher, 2006: Stochastic forcing of the North Atlantic wind-driven ocean circulation. Part II: An analysis of the dynamical ocean response using generalized stability theory. J. Phys. Oceanogr., 36, 316–334. Dobrovolski, S. G., 2000: Stochastic Climate Theory. Springer, 282 pp. Farrell, B. F., and P. Ioannou, 1993: Stochastic dynamics of baroclinic waves. J. Atmos. Sci., 50, 4044–4057. ——, and ——, 1995: Stochastic dynamics of the midlatitude atmospheric jet. J. Atmos. Sci., 52, 1642–1656.

314


——, and ——, 1996a: Generalized stability theory. Part I: Autonomous operators. J. Atmos. Sci., 53, 2025–2039. ——, and ——, 1996b: Generalized stability theory. Part II: Nonautonomous operators. J. Atmos. Sci., 53, 2041–2053. Frankignoul, C., and P. Müller, 1979: Quasi-geostrophic response of an infinite ␤-plane ocean to stochastic forcing by the atmosphere. J. Phys. Oceanogr., 9, 104–127. ——, ——, and E. Zorita, 1997: A simple model of the decadal response of the ocean to stochastic wind forcing. J. Phys. Oceanogr., 27, 1533–1546. Fu, L. L., and R. D. Smith, 1996: Global ocean circulation from satellite altimetry and high-resolution computer simulations. Bull. Amer. Meteor. Soc., 77, 2625–2636. Garzoli, S. L., and C. Simionato, 1990: Baroclinic instabilities and forced oscillations in the Brazil/Malvinas confluence front. Deep-Sea Res., 36A, 681–703. Goldstein, J., 1985: Semigroups of Linear Operators and Applications. Mathematical Monogr., Oxford University Press, 245 pp. Haidvogel, D. B., and P. B. Rhines, 1983: Waves and circulation driven by oscillating winds in an idealized ocean basin. Geophys. Astrophys. Fluid Dyn., 25, 1–63. Holland, W., 1978: The role of mesoscale eddies in the general circulation of the ocean—Numerical experiments using a wind-driven quasi-geostrophic model. J. Phys. Oceanogr., 8, 363–392. Kiehl, J., J. Hack, G. Bonan, B. Boville, B. Briegleb, D. Williamson, and P. J. Rasch, 1996: Description of the NCAR Community Climate Model (CCM3). NCAR Tech. Note NCAR/ TN-420⫹STR, 152 pp. Kleeman, R., and A. Moore, 1997: A theory for the limitations of ENSO predictability due to stochastic atmospheric transients. J. Atmos. Sci., 54, 753–767. Large, W. G., W. R. Holland, and J. C. Evans, 1991: Quasigeostrophic ocean response to real wind forcing: The effects of temporal smoothing. J. Phys. Oceanogr., 21, 998–1017. ——, J. C. McWilliams, and S. M. Doney, 1994: Oceanic vertical mixing: A review and a model with a nonlocal boundary layer parameterization. Rev. Geophys., 32, 363–403. Leetmaa, A., 1978: Fluctuating winds—Energy source for mesoscale motions. J. Geophys. Res., 83, 427–430. Lippert, A., and R. Käse, 1985: Stochastic wind forcing of baroclinic Rossby waves in the presence of a meridional boundary. J. Phys. Oceanogr., 15, 184–194. ——, and P. Müller, 1995: Direct atmospheric forcing of geostrophic eddies. J. Phys. Oceanogr., 25, 106–121. Luther, D. S., A. D. Chave, and J. H. Filloux, 1990: Evidence for local and nonlocal barotropic responses to atmospheric forcing during BEMPEX. Geophys. Res. Lett., 17, 949–952. Magaard, L., 1977: On the generation of baroclinic Rossby waves in the ocean by meteorological forces. J. Phys. Oceanogr., 7, 359–364. McWilliams, J. C., 1977: A note on a consistent quasigeostrophic model in a multiply connected domain. Dyn. Atmos. Oceans, 1, 427–441. Milliff, R. F., and A. Robinson, 1992: Structure and dynamics of the Rhodes Gyre system and dynamical interpolation for estimates of mesoscale variability. J. Phys. Oceanogr., 22, 317– 337. ——, and J. C. McWilliams, 1994: The evolution of boundary pressure in ocean basins. J. Phys. Oceanogr., 24, 1317–1338. ——, and J. Morzel, 2001: The global distribution of the time-

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average wind stress curl from NSCAT. J. Atmos. Sci., 58, 109–131. ——, W. G. Large, W. R. Holland, and J. C. McWilliams, 1996: The general circulation responses of high-resolution North Atlantic Ocean models to synthetic scatterometer winds. J. Phys. Oceanogr., 26, 1747–1768. Müller, P., 1997: Coherence maps for wind-forced quasigeostrophic flows. J. Phys. Oceanogr., 27, 1927–1936. ——, and C. Frankignoul, 1981: Direct atmospheric forcing of geostrophic eddies. J. Phys. Oceanogr., 11, 287–308. Niiler, P. P., and C. J. Koblinsky, 1989: A local time dependent Sverdrup balance in the northeast Pacific. Science, 229, 754– 756. Osychny, V., and P. Cornillon, 2004: Properties of Rossby waves in the North Atlantic estimated from satellite data. J. Phys. Oceanogr., 34, 61–76. Pedlosky, J., 1987: Geophysical Fluid Dynamics. Springer-Verlag, 710 pp. ——, 1996: Ocean Circulation Theory. Springer-Verlag, 453 pp. Philander, S. G. H., 1978: Forced oceanic waves. Rev. Geophys. Space Phys., 16, 15–46. Phillips, N. A., 1966: Large scale eddy motion in the western Atlantic. J. Geophys. Res., 71, 3883–3891. Pinardi, N., and R. F. Milliff, 1989: A note on consistent quasigeostrophic boundary conditions in partially open, simply and multiply connected domains. Dyn. Atmos. Oceans, 14, 65–67. Rhines, P. B., 1975: Waves and turbulence on a beta-plane. J. Fluid Mech., 69, 417–443. ——, 1977: The dynamics of unsteady currents. The Sea, E. D. Goldberg et al., Eds., Marine Modeling, Vol. 6, John Wiley and Sons, 189–318. Rienecker, M. M., and L. L. Ehret, 1988: Wind stress curl variability over the North Pacific from the comprehensive oceanatmosphere data set. J. Geophys. Res., 93, 5069–5077. Samelson, R. M., 1990: Evidence for wind-driven current fluctuations in the eastern North Atlantic. J. Geophys. Res., 95, 11 359–11 368. ——, and B. Shrayer, 1991: Currents forced by stochastic winds with meridionally varying amplitude. J. Geophys. Res., 96, 18 425–18 429. Schmitz, W. J., and W. R. Holland, 1986: Observed and modeled mesoscale variability. J. Geophys. Res., 91, 9624–9638. Semtner, A. J., and W. R. Holland, 1978: Intercomparison of quasi-geostrophic simulations of the western North Atlantic circulation with primitive equation results. J. Phys. Oceanogr., 8, 735–754. Spall, M. A., 1988: Regional ocean modeling: Primitive equation and quasigeostrophic studies. Ph.D. thesis, Harvard University, Harvard Reports in Meteorology and Oceanography, No. 28, 282 pp. ——, and R. Robinson, 1989: A new hybrid coordinate open ocean primitive equation model. Math. Comput. Simul., 31, 241–269. Stammer, D., and C. Wunsch, 1999: Temporal changes in the mesoscale variability of the oceans. Deep-Sea Res., 46B, 77– 108. Sura, P., and C. Penland, 2002: Sensitivity of a double-gyre ocean model to details of stochastic forcing. Ocean Modell., 4, 327– 345. Trefethen, L. N., 1997: Pseudospectra of linear operators. SIAM Rev., 39, 383–406. ——, A. E. Trefethen, S. C. Reddy, and T. A. Driscoll, 1993: Hy-

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CHHAK ET AL.

drodynamic stability without eigenvalues. Science, 261, 578– 584. Treguier, A. M., and B. L. Hua, 1987: Oceanic quasi-geostrophic turbulence forced by stochastic wind fluctuations. J. Phys. Oceanogr., 17, 397–411. Veronis, G., 1970: Effect of fluctuating wind on ocean circulation. Deep-Sea Res., 17, 421–434. ——, and H. Stommel, 1956: The action of variable wind stresses on a stratified ocean. J. Mar. Res., 15, 43–75. Wearn, R. B., and D. J. Baker, 1980: Bottom pressure measurements across the Antarctic Circumpolar Current and their relation to the wind. Deep-Sea Res., 27, 875–888.

315

Wilks, D. S., 1995: Statistical Methods in the Atmospheric Sciences: An Introduction. Academic Press, 467 pp. Willebrand, J., 1978: Temporal and spatial scales of the wind field over the North Pacific and North Atlantic. J. Phys. Oceanogr., 8, 1080–1094. ——, S. G. H. Philander, and R. C. Pacanowski, 1980: The oceanic response to large-scale atmospheric disturbances. J. Phys. Oceanogr., 10, 411–429. Wright, T. G., and L. N. Trefethen, 2001: Large-scale computation of pseudospectra using ARPACK and EIGS. SIAM J. Sci. Comput., 23, 591–605.