## 1. Introduction

At middle and high latitudes, the stochastic component of wind forcing associated with atmospheric “weather” on synoptic time scales is comparable in magnitude to the wind forcing associated with the seasonal cycle (Willebrand 1978; Chave et al. 1991; Samelson and Shrayer 1991), and observations reveal that the stochastic component exerts considerable influence on the ocean circulation (e.g., DeRycke and Rao 1973; Niiler and Koblinsky 1989; Brink 1989; Koblinsky et al. 1989; Luther et al. 1990; Samelson 1990; Chave et al. 1992). The impact of stochastic wind forcing on the large-scale ocean circulation has received considerable attention and has a long history, beginning with the work of Veronis and Stommel (1956), Phillips (1966), and Veronis (1970). Subsequent investigations were focused on the excitation of middle latitude Rossby waves by stochastic variations in wind stress forcing (e.g., Magaard 1977; Philander 1978; Frankignoul and Müller 1979; Willebrand et al. 1980; Large et al. 1991; Fu and Smith 1996; Moore 139; Sura and Penland 2002; Dewar 2003; Sirven 2005) and on the coherence between the forcing and the response (Brink 1989; Samelson 1990; Müller 1997). The impact of stochastic variations in surface heat and freshwater fluxes in concert with the wind has also been explored (e.g., Hazeleger and Drijfhout 1999; Herbaut et al. 2002), and in recent years the circulation variability on decadal time scales resulting from stochastic excitations has attracted attention (e.g., Frankignoul et al. 1997; Cessi and Louazel 2001; Yang et al. 2004; Sirven et al. 2007). Attention has also recently shifted to the Southern Ocean where stochastic wind variability is considerable (e.g., Wearn and Baker 1980; Weijer 2005; Aiken and England 2005; Weijer and Gille 2005).

The role of stochastic forcing in promoting instability and eddy formation is of particular relevance to the present study and has been explored both from a theoretical point of view (e.g., Rhines 1975; Haidvogel and Rhines 1983; Treguier and Hua 1987) and in relation to observations of the real ocean (e.g., Leetmaa 1978; Garzoli and Simionato 1990; Lippert and Müller 1995). The eddies that are generated this way can rectify the large-scale circulation forming zonal flows and enhancing western boundary currents (Alvarez et al. 1997; Seidov and Marushkevich 1992). More generally, forcing of the large-scale circulation by eddies and boundary effects can be conveniently viewed as a stochastic process (Mariano et al. 2003; Berloff 2005a,b) leading to rectified large-scale circulations.

Using generalized stability theory (Farrell and Ioannou 1996a,b), Chhak et al. (2006a,b, hereafter C2006a and C2006b, respectively) introduced a new interpretation of the stochastically induced ocean response in terms of the interference of suitably aligned nonorthogonal Rossby waves. In these experiments, C2006a and C2006b examined the response of 1° resolution quasigestrophic (QG) model of the North Atlantic Ocean to stochastic wind stress forcing and found that the response was greatest in the vicinity of the western boundary, largely the result of the interference of Rossby waves over the sloping bottom and waves arrested by the western boundary current. Following Moore (1999), C2006b also found that the spatial patterns of stochastic forcing most effective for promoting variance in the ocean circulation are typically basin scale and could project significantly onto the large-scale atmospheric teleconnection patterns and quasistationary waves that dominate the middle and high latitude atmospheric circulations in winter. One such mode of atmospheric variability that is dominant in the North Atlantic is the North Atlantic Oscillation (NAO).

The NAO, first identified by Walker and Bliss (1932), is characterized by variations in sea level pressure between the Icelandic low and Azores high. While most of the power in the NAO is at multiyear and decadal periods (the “climate regime”) and is probably associated with coupled air–sea interactions (Marshall et al. 2001), there is significant variability on weekly and monthly time scales, as well, associated with internal atmospheric variability (the “weather regime”)—in particular, wintertime standing oscillations in the Northern Hemisphere planetary wave patterns—and fluctuations in the jet stream over the western North Atlantic.

The present study extends the work of Chhak and Moore (2007, hereafter CM2007), who specifically investigated the impact of the stochastic weather regime component of the NAO on the ocean circulation. Using the same 1°-resolution QG model as C2006a, CM2007 investigated the equilibrium response of asymptotically stable subtropical and subpolar gyre circulations (i.e., circulations that do not support unstable eigenmodes) to stochastic forcing with the structure of the NAO. CM2007 found that, while the NAO weather regime can excite significant variability in the circulation, particularly near the western boundary, the coherence between the stochastic forcing and the ocean response is small, suggesting that stochastically excited circulations may be difficult to identify from observations. Here we have used an eddy-permitting QG model with ⅕° × ⅙° resolution to explore the ocean response to the stochastic component of the NAO on time scales ranging from days to months as they develop on the inherently unstable gyre circulation.

The basin-scale stochastic forcing associated with the NAO, as well as other atmospheric teleconnection patterns and quasistationary waves, projects quite naturally onto the barotropic ocean circulation (Veronis and Stommel 1956). As a result, stochastically induced variability can extend all the way to the bottom and can excite topographic Rossby waves (TRWs) in regions of sloping bathymetry (McWilliams 1974; Willebrand et al. 1980). TRW activity has been shown to contribute significantly to deep circulations in the North Atlantic (Thompson and Luyten 1976; Thompson 1977, 1978; Hogg 1981) and Gulf of Mexico (DeHaan and Sturges 2005; Hamilton 2007) and has been invoked to explain recirculation zones in the North Atlantic subtropical gyre (Hogg 1988; Malanotte-Rizzoli et al. 1995; Mizuta and Hogg 2004). More generally, as noted above, Rossby wave activity in the form of eddies can also drive recirculations (Holland and Rhines 1980; Seidov and Marushkevich 1992). Over time TRW vorticity and energy can accumulate along continental shelves, and the associated eddy Reynolds stress divergence can rectify the wave circulations and contribute to the mean flow. An important generating mechanism for TRWs are currents characterized by transient meanders such as the Gulf Stream (Hogg 1988) and the Loop Current in the Gulf of Mexico (Hamilton 2007). Our results suggest that stochastic excitation of TRWs may be another effective mechanism that can significantly influence the mean circulation via nonlinear rectification.

This paper is complementary to the insightful study by Willebrand et al. (1980), who exposed the destructive influence of bathymetry on the coherence between stochastic forcing and the ocean response. The ideas and results presented here are relevant not only to the response of the real ocean to stochastic forcing but also to the problem of ocean prediction in the presence of the inevitable errors and uncertainties that exist in the surface boundary conditions owing to the inherently unpredictable nature of the overlying atmosphere. The structure of the paper is as follows: Section 2 describes the model used, followed in section 3 by a description of the model forcing and the large-scale circulation. Section 4 explores the stochastically induced ocean response as revealed by ensembles of model solutions, and in sections 5 and 6, we apply the ideas of generalized stability theory to further understand the stochastically induced ocean response. The role of topographic Rossby waves in controlling stochastically induced variability is explored in section 7, and a summary and discussion follows in section 8.

## 2. Model description

*n*= 1,… , 5, refers to each vertical level of thickness

*H*;

_{n}*ψ*is the streamfunction; the Jacobian operator is defined by

_{n}*J*(

*ψ*,

_{n}*ζ*) = ∂

_{n}*ψ*/∂

_{n}*x*∂

*ζ*/∂

_{n}*y*− ∂

*ψ*/∂

_{n}*y*∂

*ζ*/∂

_{n}*x*;

*W*(

_{E}*t*) is the surface Ekman pumping velocity and is nonzero only in the surface layer;

*g*′

_{n}= (

*ρ*

_{n+1}−

*ρ*

_{n})

*g*/

*ρ*

_{0}are the reduced gravities of each layer of density

*ρ*;

_{n}*f*

_{0}= 2ΩsinΘ

_{0}, Ω = 7.292 × 10

^{−5}rad s

^{−1}is the angular velocity of the earth, Θ

_{0}= 30.8°N is the central latitude of the model domain;

*Z*(

_{b}*x*,

*y*) describes the bathymetry;

*δ*

_{i}_{,n}is the Kronecker delta function;

*r*= 1 × 10

^{−7}s

^{−1}is the coefficient of bottom friction; and the Laplacian diffusion coefficient

*A*750 m

_{H}=^{2}s

^{−1}where the width of the western boundary layer (

*A*/

_{H}*β*)

^{1/3}∼34 km is resolved by the model grid.

The boundary conditions imposed on (1) are free-slip (*ζ _{n}* = 0) and no-normal flow at the boundary, meaning ∂

*ψ*/∂

*s =*0 along the boundary where

*s*is the tangential direction to the boundary, and a mass conserving consistency constraint is imposed along the solid boundaries (McWilliams 1977; Pinardi and Milliff 1989; Milliff and McWilliams 1994; see also C2006a for more details). As in MLHM, full amplitude bottom slope is used, a common practice in QG models with realistic geometries (e.g., Milliff and Robinson 1992; Blayo 1994). Comparisons of primitive equation and QG models using full amplitude bathymetry have shown that they yield remarkably similar solutions (Semtner and Holland 1978; Spall 1988; Spall and Robinson 1989). The tangent linear and adjoint versions of the QG model were also used for the calculations in this paper.

## 3. Ocean winds and the large-scale ocean circulation

Several approaches can be used to isolate the stochastic component of surface forcing (e.g., Frankignoul and Müller 1979; Willebrand et al. 1980; Rienecker and Ehret 1988). Here we adopt the approach of Kleeman and Moore (1997), identifying the stochastic surface forcing as that due solely to the internal variability of the atmosphere and independent of interannual variations in SST boundary conditions. Following C2006a, we used surface winds from a 100-yr integration of the NCAR Community Climate Model, version 3 (CCM3) (G. Branstator 2004, personal communication) during which the same annual cycle of SST was imposed each year. Therefore, there are no variations in surface wind due to interannual variations in SST, unlike National Centers for Environmental Prediction (NCEP)–NCAR or European Centre for Medium-Range Weather Forecasts (ECMWF) reanalysis products in which interannual variations in SST exert considerable influence on the state of the atmosphere. The mean winds derived from the CCM3 compare favorably with the observed surface winds over the North Atlantic from the Comprehensive Ocean–Atmosphere Data Set (COADS) (not shown).

Ekman pumping velocity anomalies *w _{E}*(

*t*) were computed from CCM3 winds by removing a daily climatology so that

*w*(

_{E}*t*) is independent of variations in the SST boundary condition and due

*primarily*to the internal variability of the atmosphere, our working definition of the stochastic component of Ekman pumping velocity. The difference

*t*) =

*W*

_{E}(

*t*) −

*w*

_{E}(

*t*) represents the deterministic component of Ekman pumping velocity associated with the seasonal cycle. Based on these definitions

*w*will be dominated by variability associated with day-to-day weather variability, although it will contain energy associated with coherent quasistationary waves and the weather regimes of atmospheric teleconnection patterns.

_{E}While the NAO is present year-round (Barnston and Livezey 1987), it is most prominent during wintertime (Wallace and Gutzler 1981; Marshall et al. 2001). Our interest lies in the stochastic component of *W _{E}*(

*t*) associated with the NAO, which, following Branstator (2002), was identified by computing the December–February (DJF) variance of

*w*(

_{E}*t*). Figure 1a shows the portion of the leading EOF of

*w*associated with the NAO, hereafter denoted as

_{E}*ŵ*

_{E(NAO)}, that spans the model domain. Figure 1a exhibits the common dipole structure associated with the observed pressure anomalies over the North Atlantic (Hurrell 1995), and the time-dependent forcing

*w*

_{E}_{(NAO)}(

*t*) associated with the stochastic component of the NAO is given by

*w*

_{E(NAO)}(

*t*) =

*a*

_{NAO}(

*t*)

*ŵ*

_{E(NAO)}. The decorrelation time (

*τ*) of

_{c}*a*

_{NAO}(

*t*) was found to be ∼5 days as shown in Fig. 1b, which shows the autocorrelation function of

*a*

_{NAO}(

*t*).

The circulation resulting from (1) subject to the seasonal cycle of forcing *t*) will be denoted by *ψ*_{n}^{B}(*t*). After several decades the model energy and enstrophy reach a statistically stationary state (see MLHM). Figure 2a shows the time-averaged barotropic transport from the last 75 years of a 190-yr model integration and reveals a well-developed subtropical gyre with a maximum transport in the Gulf Stream recirculation zone of 61 Sv (Sv ≡ 10^{6} m^{3} s^{−1}). The standard deviation of the surface streamfunction *ψ*_{1}^{B}(*t*) about the monthly averaged *ψ*_{1}^{B}(*t*) during years 115 to 190 is shown in Fig. 2b and is largest near the Gulf Stream separation latitude.

MLHM note that separation of the Gulf Stream from the western boundary in the model occurs farther north than observed, a behavior symptomatic of a weak subpolar gyre impeded by the location of the closed boundary at 52°N. Following MLHM, *t*) was artificially enhanced by adding *w _{c}*(

*y*), where

*w*(

_{c}*y*) = 0 for

*y*≤ 35°N, and increases linearly to 2.1 × 10

^{−6}m s

^{−1}at the northern boundary. The correction

*w*(

_{c}*y*) is time invariant and increases the strength of the subpolar gyre circulation in the model causing the Gulf Stream to separate near Cape Hatteras. The circulation shown in Figs. 2a and 2b includes the effects of

*w*(

_{c}*y*), and the mean Gulf Stream flow near the separation point is 1.4 m s

^{−1}compared to observed values of ∼1–2 m s

^{−1}. The persistent meanders immediately downstream of Cape Hatteras in Fig. 2a are unrealistic, but do not significantly influence the conclusions of section 8.

## 4. Stochastically induced variability

When (1) is subject to an Ekman pumping velocity comprised of the seasonal cycle *t*) and the stochastic forcing *w _{E}*

_{(NAO)}(

*t*) due to the NAO, the stochastically induced perturbations increase the variance of the circulation by more than a factor of 2 when compared to the intrinsic variability of the model forced by

*t*) alone. To illustrate, Fig. 2c shows the standard deviation of (

*ψ*

_{1}

^{SF}(

*t*) −

*ψ*

_{1}

^{B}(

*t*)), where

*ψ*

_{1}

^{SF}(

*t*) denotes the model solution subject to

*t*) +

*w*

_{E(NAO)}(

*t*) for the period: year 115 to 190. Figures 2b and 2c indicate that the difference between the stochastically forced and the seasonally forced circulations is as large as the intrinsic variability and that

*w*

_{E}_{(NAO)}(

*t*) enhances the intrinsic variability of the circulation in the most dynamically active regions.

The 75-yr run of the nonlinear model subject to NAO stochastic forcing, *w _{E}*

_{(NAO)}(

*t*), yields the impact of continuously induced perturbations on the circulation variance of the resulting statistically steady state (cf. Fig. 2c). However, the ergodic nature of the system means that the variance in time of

*ψ*

^{SF}(

*t*) is equivalent to the variance of an ensemble of stochastically forced solutions at a single time. Therefore, in subsequent sections, the dynamics of variability induced by

*w*

_{E}_{(NAO)}(

*t*) will be explored using an ensemble approach. In addition, we restrict attention to the linear phase of ensemble variance development so that we may use the ideas of generalized stability theory. Two important first steps are therefore to determine (i) the minimum ensemble size that will yield meaningful statistics and (ii) the time interval during which linear dynamics are valid. These issues are addressed in the following subsections.

The ensembles presented here can be interpreted in two ways. They can be considered as representative of the range of possible circulations that may be driven in the real ocean by the weather regime component of the NAO. This view of the ensembles has important implications for the way that we interpret observations of the ocean. Alternatively, the ensembles can be considered as an indication of the uncertainty in forecasts of the ocean circulation resulting from uncertainties and errors in the atmospheric forcing due to the inherent limit of predictability of the NAO weather regime. This view of the ensembles has important implications for the limit of predictability of the ocean circulation due to uncertainties in the surface boundary conditions.

### a. Ensemble generation

*W*

_{E}(

*t*)≡

*t*) +

*w*

_{E(NAO)}(

*t*), where different realizations of

*w*

_{E}_{(NAO)}(

*t*) were used to generate an ensemble of

*M*runs, each of 30 days duration. Each ensemble member was started from the same initial condition corresponding to

*ψ*

^{B}

_{n}on 1 January, year 116 (denoted

*t*= 0) from the model forced by

*t*), the seasonal cycle of Ekman pumping. Each ensemble member was generated by forcing the model with

*t*) +

*w*

_{E(NAO)}(

*t*), where different wintertime (DJF) realizations of

*w*

_{E}_{(NAO)}(

*t*) were selected at random from the 100-yr run of CCM3 described in section 3. The resulting solutions of (1) and (2) will be denoted

*ψ*

_{n}

^{m}(

*t*) and

*ζ*

_{n}

^{m}(

*t*), where

*m*= 1, … ,

*M*refers to each ensemble member. Two measures of the variance of the ensemble about the ensemble mean were considered: (i) perturbation energyand (ii) perturbation enstrophywhere ∫∫

*o*…

*dx dy*denotes the area integral over the entire model domain; angle brackets represents the ensemble mean of the

*M*realizations; and Δ

*ζ*

_{n}(

*t*) =

*ζ*

_{n}

^{m}(

*t*) − 〈

*ζ*

_{n}(

*t*)〉 and Δ

*ψ*

_{n}(

*t*) =

*ψ*

_{n}

^{m}(

*t*) − 〈

*ψ*

_{n}(

*t*)〉. During the early stages of ensemble development of interest here, 〈

*ψ*

_{n}(

*t*)〉≡

*ψ*

_{n}

^{B}(

*t*) and 〈

*ζ*

_{n}(

*t*)〉≡

*ζ*

_{n}

^{B}(

*t*), where

*ψ*

_{n}

^{B}(

*t*) is the vorticity associated with the solution

*ζ*

_{n}

^{B}(

*t*) discussed in section 3. Here, Δ

*E*yields information about variance associated with fluctuations in geostrophic circulations and variance maintained by the release of energy from

*ζ*

_{n}

^{B}(

*t*) by barotropic and baroclinic processes, while Δ

*Q*more specifically isolates the variance associated with Rossby wave activity and vortex stretching and compression as fluid columns move relative to

*ψ*

_{n}

^{B}(

*t*) and over bathymetry.

Figures 3a and 3b show Δ*E*(*t*) and Δ*Q*(*t*) versus *M* for *t* = 1, 10, 20, and 30 days and treveal that, for *M >* 80, the ensemble variance does not vary significantly with increasing *M*. Therefore, in all experiments presented hereafter we used *M =* 80.

### b. The tangent linear model

*ψ*

_{n}

^{B}(

*t*). The TLM can be written in matrix–vector notation aswhere

*δ*

**denotes the vector of perturbation vorticity gridpoint values and**

*ζ*

_{E}_{(NAO)}(

*t*) is the vector of stochastic forcing associated with the NAO and is nonzero only in level 1:

*t*) is the discretized tangent linear operator linearized about

*ψ*

_{n}

^{B}(

*t*). Specifically,

*t*) is the discretized form of the operators in (1), where

*t*)

*δ*

*≡ −*

**ζ***J*(

*ψ*

_{n}

^{B},

*δζ*

_{n}) − (

*δψ*

_{n},

*ζ*

_{n}

^{B}) −

*δ*

_{5,n}

*J*(

*δψ*

_{5,}

*f*

_{0}

*Z*

_{b}/

*H*

_{5}) −

*β*∂

*δψ*

_{n}/∂

*x*+

*A*

_{H}∇

^{2}

*δζ*

_{n}−

*δ*

_{5,n}

*r*

*δζ*

_{5}. The perturbation streamfunction vector

*δ*is related to

**ψ***δ*according to the perturbation form of (2), which we denote in discrete form as

**ζ***δ*=

**ζ****L**

*δ*, where

**ψ****L**represents the discretized vorticity operator. Physically,

*t*) is the matrix that is applied every time step in the TLM (3) that yields the time rate of change of

*δ*

**due to advection and dissipation of QG potential vorticity.**

*ζ**M*= 80 different wintertime (DJF) realizations of

_{E}_{(NAO)}(

*t*) to generate an ensemble of TLM solutions, each linearized about

*ψ*

_{n}

^{B}(

*t*), in this case a 90-day time-evolving solution of (1) corresponding to the period 1 December year 116 to 28 February year 117. The variance of the TLM ensemble about the ensemble mean was quantified in terms of (i) perturbation energyand (ii) perturbation enstrophywhere

*δζ*and

_{n}*δψ*are solutions of (3). Time series of ln[

_{n}*δE*(

*t*)] and ln[

*δQ*(

*t*)] are shown in Figs. 4a and 4b, respectively. Figures 4a and 4b also show time series of the ensemble variances ln(Δ

*E*(

*t*)) and ln(Δ

*Q*(

*t*)) from 90-day runs of the stochastically forced nonlinear model forced with the same

*M =*80 realizations of

_{E}_{(NAO)}(

*t*) used in the TLM. The ensemble variance of the TLM and nonlinear solutions are very similar for the first 40 days. Beyond 40 days the variance of the TLM ensemble is significantly greater than that of the nonlinear model, as expected based on linear considerations. Based on Fig. 4, 40 days represents an upper bound on the time interval during which the tangent linear assumption inherent in (3) is a good approximation for wintertime forcing. By day 30 (90), the nonlinear model ensemble variance is ∼2% (10%) of the level of asymptotic stochastically induced variance of

*ψ*

^{SF}(

*t*) described earlier (cf. Fig. 2c).

### c. The autonomous versus nonautonomous case

*t*,

*τ*) is an operator called the propagator and advances unforced solutions of (3) from time

*t*to time

*τ*. The second term on the rhs of (4) is the particular integral in the presence of stochastic forcing

*a*

_{NAO}(

*t*)

_{E(NAO)}. The form of

*t*,

*τ*) depends on

*t*) [defined in relation to (3)] and the time evolution of the circulation

*ψ*

_{n}

^{B}(

*t*). If

*ψ*

_{n}

^{B}describes a steady circulation, then

*t,τ*) =

*e*

^{(τ−t)}.

_{E}_{(NAO)}(

*t*)〉

*=*〈

*a*

_{NAO}(

*t*)〉

_{E(NAO)}= 0, and each ensemble member has the same initial condition,

*δ*(0) = 0, in which case the ensemble mean of the perturbations 〈

**ζ***δ*

**(**

*ζ**t*)〉 = 0. Therefore, by definition, the ensemble variance

*V*(

*t*) about the ensemble mean at time

*t*, is given bywhere

*t*) is the covariance matrix of the stochastically forced ensemble given bywhich follows from (4), given that

*δ*

**(0) = 0 for each ensemble member;**

*ζ**α*

^{2}is the variance of the NAO principal component time series

*a*

_{NAO}(

*t*);

*g*(

*t*′,

*t*″) is the wintertime autocorrelation of

*a*

_{NAO}(

*t*), shown in Fig. 1b; and superscript T denotes the transpose. The second equality in (5) arises by identifying the variance as the trace of the ensemble covariance matrix

*V = δE*) and enstrophy (

*V = δQ*) norms. As noted in section 4a,

*δE*provides information about the role of barotropic and baroclinic processes in controlling the ensemble variance, while

*δQ*highlights the nature of variance associated with Rossby wave activity. For the enstrophy norm

*υ*

_{i}), a diagonal matrix comprised of the volumes

*υ*of the model grid cells, while for the energy norm

_{i}*υ*

_{i})

^{−1}, where

The time dependence of *t*) in (3) results from the time evolution of *ψ*_{n}^{B}(*t*), and (3) describes a nonautonomous system. Some analyses in sections 5 and 6 are more easily facilitated by considering (3) linearized about the time-mean _{n}^{B} of the circulation *ψ*_{n}^{B}(*t*) (e.g., Fig. 2a). In this case *t*) is computationally demanding to compute directly. However, for the autonomous case *t*_{1}, *t*_{2}) = *e*^{(t2−t1)}, and *t*) simplifies to *t*) = ∫_{0}^{t}∫_{0}^{t} *g*(*t*′, *t*″) **d**(*t*′) **d**^{T} (*t*″) *dt*′ *dt*″, where **d**(*τ*) = (*α**f*_{0}/*H*_{1})^{1/2} *e*^{(t − τ)}_{E(NAO)}. The eigenvectors (EOFs) of *t*) for each norm in this case can be computed very easily, and the leading EOFs are indicative of the spatial variance associated with each norm. The EOF spectrum of *t*) defined relative to *δE* and *δQ* is dominated by the leading member, which is shown in Fig. 5 for both variance norms for the surface streamfunction (Figs. 5a and 5b) and deep ocean streamfunction (Figs. 5c and 5d) on day 30. For *δQ*(*δE*), the leading EOF accounts for 77% (72%) of the stochastically induced variance. At the surface (Figs. 5a and 5b) the large-scale spatial structure of *δψ* for both norms is similar to the NAO EOF structure in Fig. 1a and describes a large-scale modulation of the subtropical–subpolar gyre circulation. Superimposed on this are smaller-scale closed circulations in the vicinity of the western boundary downstream of Cape Hatteras. In the deep ocean the variance *δQ* takes the form of deep currents primarily along the western margins of the abyssal plains as shown in Fig. 5c. For *δE* the deep ocean circulations mirror those of the surface but are of opposite sign and weaker (Fig. 5d).

The remaining 23% of *δQ* and 28% of *δE* are explained by smaller-scale circulations as illustrated in Figs. 5e and 5f, which show the second EOF for each norm for the surface streamfunction. The remaining significant EOFs (not shown) have qualitatively similar structures. In general, we find that the variance *δQ* is explained by more smaller-scale structures near the western boundary than the variance *δE*.

For the autonomous case the ensemble variance in (5) is also given by *V*(*t*) = ∫_{0}^{t}∫_{0}^{t}*g*(*t*′, *t*″) **d**^{T}(*t*′) **d** (*t*″) *dt*′ *dt*″, plotted in Fig. 4 for *δQ* and *δE* (dashed curve). A comparison of Fig. 4 with the variance computed directly from an *M* = 80 ensemble (not shown) matches very well the theoretical value of (5), further indicating that *M* = 80 is sufficient. Figure 4 shows that the evolution of the TLM ensemble variance *δE* and *δQ* is qualitatively similar, irrespective of whether (3) is linearized about *ψ*_{n}^{B}(*t*) or _{n}^{B}.

### d. The stochastically induced ocean response

*δE*and

*δQ*can be expressed symbolically asExplicit mathematical expressions for each term in (6) and (7) are presented in Appendix A and represent sources and sinks of energy and enstrophy variance arising from various processes, including

*E*

_{trop}and

*Q*

_{relative}, that are proportional to the perturbation momentum stresses associated with the straining component of

*ψ*

_{n}

^{B}(

*t*);

*E*

_{clin}, arising from interactions of

*δψ*with the vertical shear of

_{n}*ψ*

_{n}

^{B}(

*t*);

*E*

_{force}and

*Q*

_{force}, due to the NAO stochastic forcing

_{E}_{(NAO)}; the sinks, denoted

*E*

_{diss}and

*Q*

_{diss}, due to lateral diffusion and bottom friction;

*Q*

_{stretch}, due to perturbations that are stretched or compressed as they move relative to the deformed pressure surfaces of

*ψ*

_{n}

^{B}(

*t*); the source

*Q*

_{bathy}, due to the stretching or compression of perturbations as they move over the bottom bathymetry; and net sources and sinks of enstrophy variance at continental boundaries, denoted

*Q*

_{bndy}, due to Rossby wave reflections.

Time series of each term in (6) and (7) computed from an 80-member, 30-day ensemble of TLM solutions are shown in Fig. 6 for the nonautonomous and autonomous cases. For the nonautonomous case (Fig. 6a), the largest contributor to the growth of *δE* is *E*_{force}. The contribution of *E*_{clin} is also significant but is by and large offset by variance dissipation *E*_{diss}. The largest source of enstrophy variance growth beyond day 20 (Fig. 6b) is *Q*_{stretch} closely matched by *Q*_{relative} beyond day 25, but these are offset by variance dissipation *Q*_{diss}. Here *Q*_{bathy} and *Q*_{force} account for the remaining enstrophy growth.

In the autonomous case, the largest contribution to the growth of *δE* (Fig. 6c) is also *E*_{force}, while for *δQ* (Fig. 6d) the largest contributor to growth of variance is *Q*_{bathy}. The dominant role played by *Q*_{bathy} is evident in the deep ocean structure of the leading EOF of *δQ* in Fig. 5c. Figure 6 reveals that the balance of terms for *δE* and *δQ* variance is quite different in the autonomous and nonautonomous cases. In particular, the growth of *δE* by baroclinic processes (*E*_{clin}) is an order of magnitude larger in the latter because time averaging of *ψ*_{n}^{B}(*t*) reduces the local straining rates, particularly in the vicinity of the Gulf Stream. Nonetheless, Fig. 6a shows that the growth of variance by baroclinic processes (*E*_{clin}) in the nonautonomous case is largely offset by the loss of variance due to dissipation (*E*_{diss}).

## 5. Modal versus nonmodal growth

Much of the stochastically induced variance *δQ* of the asymptotically stable North Atlantic circulations explored by C2006a, C2006b, and CM2007 could be interepreted as linear interference of particular eigenmodes of the gyre circulation. The eigenmodes are the eigenvectors of the operator *ψ*_{n}^{B}. Both

The size of stochastically excited perturbations as they evolve in time is measured using a norm (e.g., energy and enstrophy). We will demonstrate that, in the presence of an unstable gyre circulation, the evolution of a norm is generally characterized by three phases of development: (i) a linear nonmodal phase during which rapid growth of a norm occurs (“super exponential” growth in the case of an autonomous system); (ii) a linear modal phase in which norm growth is governed by the most unstable eigenmodes; and (iii) a nonlinear phase in which norm growth is slower than in the linear phase or arrested completely. The mechanism underlying the nonmodal growth phase will be discussed in section 7.

The eigenspectrum of *ψ*_{n}^{B}(*t*), the eigenmodes of the nonautonomous system described by *t*), are not unique and depend on the circulation at time *t*.

_{n}

^{B}. Each time step

_{E}_{(NAO)}introduces a perturbation in the TLM that projects on the eigenmodes of the autonomous matrix

*V*(

*t*) at time

*t*can be expressed aswhere

**ζ̂**_{i}is the

*i*th eigenmode of

*σ*;

_{i}*c*= (

_{i}*f*

_{0}/

*H*

_{1})

**ζ**_{i}

^{†H}

_{E(NAO)}is the projection of the NAO EOF on

**ζ̂**_{i}, and

**ζ**_{i}

^{†}is the (adjoint) eigenmode of

^{−1}

^{T}

*σ*

_{i}*;

*I*represents the total number of eigenmodes of

*α*,

*g*(

*t*′,

*t*″) and

**ζ̂**_{i}

^{H}

**ζ̂**_{j}for

*i*≠

*j*in (8) represent the contribution to

*V*(

*t*) of the linear interference of the nonorthogonal eigenmodes

*. For a normal circulation*

**ζ̂**

**ζ̂**_{i}

^{H}

**ζ̂**_{j}=

*δ*

_{i, j,}, and (8) indicates that

*V*(

*t*) = ∑

_{i = 1}

^{I}

*V*

_{i}(

*t*), the sum of the variances,

*V*(

_{i}*t*), associated with each individual orthogonal mode, where

*V*(

_{i}*t*) ≤

*V*(

*t*). The ensemble variance of a normal circulation with the same eigenspectrum

*σ*and projection coefficients

*c*as

_{i}*V*(

*t*) in the nonnormal system that exceeds that expected in the equivalent normal system must, by necessity, be due to nonmodal growth. Using the equivalent normal system as a reference, an upper bound on the modal growth of

*V*(

*t*) can be derived in terms of

*V*

_{1}(

*t*), whereis the variance associated with the most unstable eigenmode

**ζ̂**_{1}.

The factor by which the ensemble variance increases over the interval [*t*_{1}, *t*] is given by *μ*(*t*) *= V*(*t*)/*V*(*t*_{1}). For the equivalent normal system, *μ*_{normal} *=*Σ* _{i}V_{i}*(

*t*)/Σ

*(*

_{i}V_{i}*t*

_{1}). If there are at least two exponentially growing eigenmodes, then

*μ*

_{1}(

*t*)

*= V*

_{1}(

*t*)/

*V*

_{1}(

*t*

_{1}) ≥

*μ*

_{normal}, and

*μ*

_{1}(

*t*) represents an upper bound on the modal growth of

*V*(

*t*) associated with the exponential growth of unstable normal (orthogonal) eigenmodes alone. Any growth in excess of

*μ*

_{1}(

*t*) must therefore be due to

*nonmodal*growth associated with the linear interference of nonnormal eigenmodes.

For the time-mean flow _{n}^{B}, *σ*_{1} and |*c*_{1}|^{2} were estimated for each norm by initializing the TLM with a unit impulse forcing with the spatial structure of _{E}_{(NAO)}, then integrating (3) for 50 years during which time the most unstable, exponentially growing eigenmode emerges. A least squares best fit applied to the resulting time series of perturbation energy and enstrophy reveals an *e*-folding growth time 1/Re(*σ*_{1}) = 274 days and a period 2*π*/Im(*σ*_{1}) = 7900 days. The associated uncertainty in this estimate for *σ*_{1} and for the estimates of |*c*_{1}|^{2} is less than 0.1%. The most unstable eigenmode takes the form of a shear instability centered near 22°N aligned with the westward branch of the subtropical gyre circulation between 55° and 75°W (not shown).

Figures 7a and 7b show time series of the growth factor ratio *μ*(*t*)/*μ*_{1}(*t*) *= V*(*t*)*V*_{1}(*t*_{1})/[*V*(*t*_{1})*V*_{1}(*t*)] for the *δQ* and *δE* norms, respectively, for *t* ≥ *t*_{1} for the stochastically forced autonomous TLM analyzed in section 4. The growth factor ratios are shown referenced to four different values of *t*_{1} = 1, 10, 20, and 30 days and are indicators of the tendency for nonmodal growth over different time intervals.

Focusing first on *δQ*, Fig. 7a reveals that when *t*_{1} *=* 1 day, then *μ*(*t*) *> μ*_{1}(*t*), and the growth of *δQ* in excess of *μ*_{1} is due to nonmodal growth as a result of the constructive linear interference of the nonorthogonal eigenmodes of *δQ* variance is only evident during the first 10 days of the ensemble. For *t*_{1} ≥ 10 days, there is no obvious evidence of nonmodal growth, and the growth factor ratios *μ*(*t*)/*μ*_{1}(*t*) are less than 1.

The symptoms of eigenmode interference on ensemble enstrophy are illustrated in Fig. 7c, which shows time series of log_{10}(*δQ*(*t*)) computed from the autonomous ensemble and log_{10}(*δQ*_{1}(*t*)) computed from (9). Here *δQ*_{1}(*t*) represents the ensemble enstrophy associated with the most unstable eigenmode in the equivalent normal system. If the circulation _{n}^{B} is associated with a normal spectrum, then *δQ*_{1}(*t*) must necessarily be less than *δQ*(*t*). Figure 7c indicates that this is not the case, meaning that the eigenmodes of *δQ*(*t*) < *δQ*_{1}(*t*). Even though the analysis of Fig. 7 is not possible for a nonautonomous *t*) in (3), nonmodal growth will enhance *δQ* in this case as well as demonstrated in section 7.

For *δE*, Fig. 7b shows that *μ*(*t*) < *μ*_{1}(*t*) for all *t*_{1}, suggesting that nonmodal growth of ensemble perturbation energy is less important. Note that *μ*_{1}(*t*) represents only an upper bound due to unstable modal growth. Therefore, the condition *μ*(*t*) < *μ*_{1}(*t*) does not demonstrate the absence of nonmodal growth, only that it cannot be identified using *μ*_{1} as a lone indicator. Time series of log_{10}[*δE*(*t*)] (Fig. 7d) computed from the autonomous ensemble and log_{10}<*δE*_{1}(*t*)> computed from (9) reveal that during the first 10 days *δE*_{1}(*t*) < *δE*(*t*), suggesting that the growth of more than one eigenmode contributes to *δE*(*t*). We will argue in section 7 that nonmodal growth of *δE* is inhibited by the large-scale nature of the NAO-induced perturbations.

## 6. Generalized stability theory

Having established that both modal and nonmodal growth of stochastically induced perturbations are important for maintaining ensemble variance *δQ*, we have applied generalized stability theory (Farrell and Ioannou 1996a) to explore the problem further. For unforced initial value problems [*a*_{NAO}(*t*) *=* 0], nonmodal growth of *δ ζ* given by (4) can be quantified by the leading singular value of

*τ*) (Moore and Farrell 1993). The leading left singular vector is the perturbation that in the linear limit, maximizes the growth of the L2-norm ‖

*τ*)‖

_{2}over the interval

*t*= [0,

*τ*]. In the case of stochastically forced circulations (

*a*

_{NAO}(

*t*) ≠ 0), the contribution of nonmodal influences on the ensemble variance can be quantified by an extension of singular value decomposition applied to the second term on the rhs of (4), as follows.

*V*(

*t*) given by (5) can be expressed alternatively aswhere

**Z**(

*t*) = (

*f*

_{0}

^{2}/

*H*

_{1}

^{2}) ∫

_{0}

^{t}∫

_{0}

^{t}

*g*(

*t*′,

*t*″)

^{T}(

*t*′,

*t*)

*t*″,

*t*)

*dt*′

*dt*″ represents the integrated influence on

*δ*

**of the stochastic forcing**

*ζ**(*

_{E}*t*) by the TLM dynamics, and (10) applies to both autonomous and nonautonomous cases.

**s**

*of*

_{j}**Z**(

*t*) are called the stochastic optimals (SOs)(Farrell and Ioannou 1996a) and represent the spatial patterns of Ekman pumping velocity that account for different fractions of the stochastically induced ensemble variance

*V*(

*t*). Expanding the symmetric matrix

*t*) as

^{T}, where

**s**

*) is the matrix of orthonormal SOs and*

_{j}**Λ**is the diagonal matrix of SO eigenvalues

*λ*

*, thenwhere*

_{j}*N =*86 260 is the total number of SOs for the QG model used here. Equation (11) shows that the ensemble variance

*δE*(

*t*) and

*δQ*(

*t*) is proportional to the projection of the NAO EOF

_{E(NAO)}(Fig. 1a) onto each SO of computed using the appropriate

The SOs can be computed using the Lanczos algorithm without explicitly computing the matrix *t*), although the computations for the nonautonomous case are prohibitively expensive at this time. However, for the autonomous case calculation of the leading members of the SO spectrum is tractable as discussed in Appendix B. For this reason we restrict our attention to the SOs of the autonomous case.

Just as the singular vectors of *τ*) in (4) represent the initial time perturbations that yield the largest growth of the L2-norm, so the SOs are the impulse forcings that when applied each time instant yield the largest variance in the L2-norm at some future time. Both cases are subject to strong control by nonmodal growth. To explore further the influence of nonmodal growth on *V*(*t*), we first isolate the dominant SOs. In the following, we consider *t*) computed for *t =* 10, 20, 30, 60, and 90 days for both the energy norm and enstrophy norm for the autonomous case linearized about _{n}^{B}.

*G*of the stochastically induced variance

_{i}*V*(

*t*) described by

*δE*(

*t*) or

*δQ*(

*t*), given byComputation of the SOs,

**s**

*, is computationally demanding, and we are restricted to considering the first 100 members of the spectrum. Figure 8 shows the cumulative stochastically induced percentage variance ∑*

_{i}_{i = 1}

^{N}

*G*explained by the leading

_{i}*N =*1, 2,…, 100 SOs on days 10, 20, and 30 for both norms. Also shown are the cumulative explained variances on days 60 and 90 based on 50 and 30 members of the SO spectrum, respectively, fewer members being computed in these cases because of the high computational cost involved. Figure 8 indicates that the first 100 members of the SO spectrum (0.1% of the entire spectrum) account for 93% and 84%, respectively, of the variability

*δE*and

*δQ*on day 10. The percentage explained variance decreases with increasing time but is still 87% for

*δE*and 79% for

*δQ*by day 30. Although the tangent linear assumption is violated beyond about day 40 for wintertime NAO forcing amplitudes (cf. Fig. 4), it will be valid for longer time intervals during other seasons when the NAO amplitude is significantly lower. Therefore, the 60- and 90-day cases can be viewed as valid during seasons other than winter.

In each case, much of the variance is captured by the first member of the SO spectrum. This is the SO that best matches the structure of _{E(NAO)} (cf. Fig. 1a). To illustrate, **s**_{1}^{E} is the SO for *δE* that accounts for the largest fraction of variance on day 10 (∼63%). The wind stress curl associated with **s**_{1}^{E} is shown in Fig. 9a and spans the entire basin. Since the SOs form an orthonormal set, the wind forcing can be expressed as *E* or *Q*. Fig. 9a shows *α**a*_{1}^{E}**s**_{1}^{E}, where *a*_{1}^{E} = *f*_{0}ρ_{0}(**s**_{1}^{E})^{T} _{E(NAO)}, and represents a typical wind stress curl pattern associated with **s**_{1}^{E}. For the enstrophy norm, **s**_{1}^{Q} is the SO that accounts for the largest fraction of ensemble variance *δQ* on day 10 (∼67%). Figure 9b shows *α**a*_{1}^{Q}**s**_{1}^{Q}, where *a*_{1}^{Q} = *f*_{0}ρ_{0}(**s**_{1}^{Q})^{T} _{E(NAO)} and is similar in structure to **s**_{1}^{E}. Also shown in Fig. 9 are the second and third members of the SO spectra for the *δE* and *δQ* variance norms on day 10. Collectively, the first three SOs account for 78% of *δE* and 76% of *δQ*.

Figure 8 indicates that, as *t* increases, the ensemble variance explained by the leading members of the SO spectrum decreases. This is because, as time increases, the leading SOs are dominated more and more by small-scale features, mostly confined to the western boundary. This is particularly true for the SOs of *δQ*. As a result the projection of _{E(NAO)} on the leading SOs [i.e., _{E(NAO)}^{T}**s**_{i} in (12)] decreases with increasing *t*.

The large-scale SO structures are controlled by the bathymetry, and comparison of Fig. 9 with Fig. 2d reveals the influence of the Mid-Atlantic Ridge and the Antilles island rise on the SO structures. In addition, further analysis (not presented) shows that the large- (small) scale SOs favor a barotropic (baroclinic) response, in agreement with Veronis and Stommel (1956), Phillips (1966), Veronis (1970), and Willebrand et al. (1980). Additional experiments (not shown) reveal that the small-scale structures evident in the SOs are dictated by the ocean circulation.

## 7. Nonnormal effects and topographic Rossby wave generation

Consider again the stochastically forced TLM (3). At each time step, a perturbation with the structure of _{E(NAO)} will be introduced into the surface layer of the model: it is the time-integrated effect of these perturbations that accounts for the stochastically induced variance of the circulation (cf. Fig. 4). Clearly, nonmodal effects can elevate the variance above that expected due to modal growth alone. Figure 10a shows a time series of the *perturbation* enstrophy growth factor *Q*(*t*)/*Q*(0) associated with a surface layer perturbation with the initial structure of _{E(NAO)} at *t* = 0 for the autonomous case. During the first 10 days or so *Q*(*t*) grows rapidly to ∼9*Q*(0). Beyond about day 10, *Q*(*t*) slowly decreases, then increases again around day 60. The behavior of *Q*(*t*) in Fig. 10a for an NAO-induced perturbation is consistent with the nonmodal growth of *δQ* in Fig. 7a. The energy of the perturbation, on the other hand, decreases monotonically over time (not shown).

*perturbation*energy density

*k*is the wavenumber and, without loss of generality, an infinite domain is assumed. The wavenumber associated with the first moment is given by

*Q*and

*E*. As the coefficients 1/

*E*and

*Q/E*

^{2}are always positive, then for perturbations dominated by perturbation enstrophy growth (∂

*Q*/∂

*t*> 0 and ∂

*Q*/∂

*t*≫ (

*Q*/

*E*)∂

*E*/∂

*t*), the mean wavenumber will increase over time; the scale of the perturbations decreases as enstrophy becomes concentrated where gradients in velocity are large. Conversely, for perturbations dominated by perturbation energy growth (∂

*E*/∂

*t*> 0 and ∂

*Q*/∂

*t*≪ (

*Q*/

*E*)∂

*E*/∂

*t*) the mean wavenumber will decrease as the perturbation evolves; the perturbation scale increases as energy moves to larger-scale energetic geostrophic circulations.

Equation (13) is valid in a bounded domain also and applies to the barotropic mode and each baroclinic mode. The sources of perturbation enstrophy and perturbation energy that control ∂*Q*/∂*t* and ∂*E/*∂*t* for the unforced initial value problem considered in Fig. 10 were discussed in relation to Eqs. (6) and (7) for a bounded ocean.

Equation (13) suggests that the basin-scale nonmodal perturbations that are stochastically excited by the NAO increase their perturbation enstrophy by decreasing their scale. On the other hand, the ability of NAO-scale nonmodal perturbations to increase their perturbation energy will be limited by the physical size of the basin; the initial perturbations already fill much of the basin, so they cannot increase their horizontal scale further and hence cannot undergo nonmodal energy growth.

The analyses of section 6 indicate that a significant fraction of *δQ* can be accounted for by a few preferred forcing patterns, the SOs. Figure 10a shows time series of *Q*(*t*)/*Q*(0) for surface perturbation introduced by **s**_{1}^{Q}, the SO that most effectively spans *δQ* at time *τ* = 10, 20, and 30 days. These SOs all have very similar large-scale structures, and perturbations associated with each of them grow rapidly, achieving an order of magnitude growth in *Q* after ∼10–15 days. Figure 10b shows the case for the dominant SOs for *τ* = 60 and 90 days. Although the tangent linear assumption is violated over these time intervals for wintertime forcing amplitudes (cf. Fig. 4), it is interesting to note that these perturbations also undergo the most rapid amplification during the first 10 days or so. Rapid growth of perturbation enstrophy over the first 10 days is a ubiquitous feature of other SOs (not shown).

Nonmodal growth of perturbation enstrophy during the first 10–15 days is associated primarily with bathymetry. This is illustrated in Fig. 10c, showing how a surface perturbation introduced by the NAO EOF evolves in the autonomous TLM for (i) the annual mean basic state, (ii) an ocean at rest, and (iii) an ocean at rest with a flat bottom. Figure 10c indicates that the most rapid growth of *Q*(*t*) is associated with the presence of bathymetry, consistent with the budget analysis of Fig. 6d. Figure 10c also indicates that inhomogeneities in _{n}^{B} account for most of the perturbation growth at times greater than 40 days.

A more dramatic illustration of the impact of *ψ*_{n}^{B}(*t*) on the evolution of NAO-induced surface perturbations is shown in Fig. 10c for the nonautonomous TLM. Rapid growth of *Q* occurs during the first 10 days, continuing through day 90. Figure 10c is an illustration of how the layer thickness variations and tighter circulation gradients of *ψ*_{n}^{B}(*t*) enhance the nonnormal growth of perturbations via *Q*_{stretch} and *Q*_{relative} in (7).

The nonmodal growth of ensemble enstrophy evident in Figs. 7a and 10 appears to be a combination of two effects. First, the NAO induces large-scale perturbations in the surface ocean that excite a predominantly barotropic response that propagates westward in the form of long Rossby waves. Second, when these large-scale waves encounter bathymetry, vortex stretching over the ridges separating the abyssal plains and the continental rise along the western boundary excite topographic Rossby waves, which are evident in the leading EOF of *δQ* in Fig. 5c. Accumulation of enstrophy in the shelf- and ridge-scale TRWs leads to the rapid growth of *Q* in Fig. 10 and accounts for the significant nonmodal growth of ensemble variance illustrated in Fig. 7a on time scales ∼10 days.

Perturbations introduced by the NAO can be viewed as a linear superposition of the nonorthogonal eigenmodes of *σ _{n}*, the relative superposition of each eigenmode will change over time to reveal the large amplitude TRWs, resulting in a growth of enstrophy. This type of nonmodal perturbation growth can arise from superposition of eigenmodes that have similar frequencies and disparate growth or decay rates, or similar growth/decay rates and disparate frequencies.

As discussed in section 1, enstrophy and energy accumulation from TRWs along continental shelves can rectify the mean circulation. To explore this idea further, the last 75 years of the nonlinear model integrations *ψ*_{n}^{B}(*t*) and *ψ*_{n}^{SF}(*t*) discussed in sections 3 and 4 were examined. Figure 12a shows the mean wintertime barotropic transport difference between the two runs, while Fig. 12b shows the mean wintertime transport difference in the deepest layer only. Recall that wintertime is when the NAO has largest amplitude. Significant differences in barotropic transport of up to 5 Sv are evident in Fig. 12a along much of the western boundary. Figure 12b reveals that stochastic forcing by the NAO also drives significant deep transports, ∼2 Sv. West of the Mid-Atlantic Ridge, many of the circulation features in Fig. 12b are correlated with the bathymetry (Fig. 2d), and some are very reminiscent of the EOF shown in Fig. 5c, suggesting that linear TRWs are the precursors that rectify the circulation over time.

Rectification of stochastically induced perturbations is due to nonlinearity and the *β* effect (planetary and topographic *β*) and can occur in several ways (see Berloff 2005a for a recent discussion). In all cases, the divergence of both the perturbation vorticity flux due to eddy stresses and perturbation potential thickness plays a key role (Holland and Rhines 1980; Alvarez et al. 1997; Mizuta and Hogg 2004). The precise nature of the process leading to the rectified circulations in Fig. 12 is unclear, but the topographic *β* effect is clearly important, and there is considerable literature on this subject, (e.g., Salmon et al. 1976; Bretherton and Haidvogel 1976; Rhines 1977; Holloway 1978).

## 8. Summary and conclusions

We have explored the influence of stochastic variations in large-scale coherent atmospheric patterns of surface forcing on the ocean circulation. In particular, we considered stochastic forcing associated with the NAO and found that the stochastically induced variability was as large as the intrinsic variability of the circulation. Our results, however, are probably very general and applicable to forcing by large-scale atmospheric teleconnection patterns and quasistationary waves in other ocean basins.

Stochastically induced perturbation growth and ensemble variance induced by the NAO are maintained both by modal and nonmodal influences depending on the choice of norm. Two different, physically motivated norms were considered as measures of stochastically induced variability: 1) perturbation energy to highlight stochastically induced changes in geostrophic circulations (e.g., Fig. 5b) and the role played by stochastically excited perturbations in liberating energy from the underlying circulation via barotropic and baroclinic energy conversion and 2) perturbation enstrophy to highlight the role played by Rossby wave dynamics (e.g., Figure 5c). For both norms the stochastically induced circulation comprised a predominantly large-scale response near the surface associated with fast barotropic Rossby waves, although a significant fraction of the response (>20%) in both norms was associated with small-scale waves. In contrast, for the enstrophy norm the deep ocean response is dominated by TRWs and bottom-trapped circulations. The nonnormality of the TRW modes enhances the stochastically induced enstrophy variance of the circulation due to the nonmodal growth of perturbations over periods ∼10 days. Furthermore, the circulation associated with the stochastically induced TRWs is rectified to yield small, but nonetheless significant, contributions to the western boundary current recirculation zones and deep flows around the margins of the abyssal plains.

For enstrophy variance, the following progression of perturbation and variance development was found: (i) a linear nonmodal phase characterized by the growth of perturbations due primarily to linear wave interference lasting ∼10–14 days; (ii) a linear modal phase characterized by growth of unstable modes, lasting ∼25–30 days; (iii) a nonlinear phase developing after about 40 days in winter (probably longer at other times of the year) when the growth of perturbations slows and probably eventually halts, as evidenced by the tendency for ensemble variance to asympotote to a constant value. In the case of energy variance, the nonmodal phase appears less important, being limited by the basin-scale nature of the stochastically excited perturbations.

Generalized stability theory reveals that the large-scale forcing associated with the NAO is relatively optimal for exciting ocean Rossby wave variability on time scales on the order of weeks, as evidenced by the fact that the dominant members of the SO spectrum explain a large fraction of the stochastically induced variance. Specfically, the first three dominant SOs collectively account for 78% of *δE* and 76% of *δQ* on day 10. Beyond about a month, the tangent linear assumption breaks down for observed wintertime forcing amplitudes but, nonetheless, is valid for smaller amplitude forcing expected at other times of the year.

Our results are relevant to the real ocean in two ways. First, they suggest that stochastic forcing can significantly influence the ocean circulation (cf. Fig. 2c), which has important ramifications for how we interpret observations. Second, seasonal predictions of the ocean circulation are likely to be significantly influenced by natural variability of the surface forcing associated with the large-scale atmospheric teleconnection patterns. However, the deterministic limit of predictability of these aspects of the surface forcing is known to be only ∼10–14 days (Molteni et al. 1993), indicating that an ensemble approach to ocean forecasting that accounts for uncertainties in ocean surface forcing, arising from the limits of atmospheric predictability and forcing errors, is essential for ocean forecasting beyond a few days. In addition, depending on the norm that is used to assess forecast error growth and forecast skill, nonmodal growth of forecast errors and forecast ensemble variance may be significant and largest during the early stages of the forecast.

Several caveats to our results and conclusions should be mentioned. First, the model considered here is QG and it is likely that ageostrophic effects not described by the model (e.g., diabatic processes and coastal circulations) may introduce additional effects not considered here. Second, some aspects of the model circulation are unrealistic, such as artificial boundaries at 9° and 52°N and the standing meanders in the vicinity of Cape Hatteras. We do not believe that changing these features will quantitatively alter the general conclusions presented here with regard to the importance of stochastic forcing and the role played by topographic Rossby waves. In fact, the influence of bathymetry on stochastically induced variance reported here agrees in many respects with the findings of Willebrand et al. (1980), who used a primitive equation ocean general circulation model. The analyses presented here, however, provide an alternative geometric interpretation of the influence of bathymetry on the stochastically induced circulation variance.

## Acknowledgments

The authors are indebted to Dr. Grant Branstator at NCAR for generously providing us with the CCM3 surface wind data. The research described here was supported by a grant from the National Science Foundation Physical Oceanography Program (OCE-0002370). 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.

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## APPENDIX A

### The Ensemble Mean Perturbation Energy and Enstrophy Equations

*δE*can be obtained by considering the ensemble mean of the dot product of the TLM (3) with − (1/2)

*δ*

**, which yieldswhere for each ensemble member**

*ψ**u*= −∂

_{n}*δψ*/∂

_{n}*y*,

*υ*= ∂

_{n}*δψ*/∂

_{n}*x*, and

*w*= ∂

_{n}*δψ*/∂

_{n}*z*;

*U*= − ∂

_{n}*ψ*

_{n}

^{B}/∂

*y*and

*V*= ∂

_{n}*ψ*

_{n}

^{B}/∂

*x*;

*γ*

_{n}

^{B}= (

*ψ*

_{n}

^{B}−

*ψ*

_{n − 1}

^{B})/

*g*′

_{n − 1}− (

*ψ*

_{n + 1}

^{B}−

*ψ*

_{n}

^{B})/

*g*′

_{n}for

*n*= 2, 3, 4, while

*γ*

_{1}

^{B}= (

*ψ*

_{1}

^{B}−

*ψ*

_{2}

^{B})/

*g*′

_{1}, and

*γ*

_{5}

^{B}= (

*ψ*

_{5}

^{B}−

*ψ*

_{4}

^{B})/

*g*′

_{4}; and ∮

_{C}indicates a counterclockwise line integral around the ocean boundary

*C*. Similarly, the ensemble mean of the dot product of (3) with (1/2)

*δ*yields an evolution equation for the ensemble mean variance in terms of

**ζ***δQ*:

## APPENDIX B

### Stochastic Optimals

*t*′,

*t*) =

*e*

^{(t−t′)}in which case

**Z**(

*t*) = (

*f*

_{0}

^{2}/

*H*

_{1}

^{2})∫

_{0}

^{t}∫

_{0}

^{t}

*g*(

*t*′,

*t*″)

*e*

^{ T(t − t′)}

*e*

^{(t − t″)}

*dt*′

*dt*″. The eigenvectors of

*t*) can be computed using the Lanczos algorithm (Golub and van Loan 1989) without explicitly computing the matrix

*t*). All that is required is the ability to compute the action of

**Z**(

*t*) on an arbitrary vector

**u**. The resulting vector

**v**=

*t*)

**u**can be written aswhere

*δ*(

**ζ***τ*) =

*e*

^{τ}

**u**is the solution of the unforced TLM (3) with initial condition

**u**. Equation (B1) for

**v**represents the solution of the adjoint of (3) (with respect to the L2-norm) forced by ∫

_{0}

^{t}

*g*(

*t*′,

*t*″)

*(*

**ζ***t*−

*t*″)

*dt*″. Therefore the action of

*t*) on an arbitrary vector can be evaluated by a single integration of the TLM followed by a single integration of the adjoint model appropriately forced. For the autonomous case, computation of the first 100 members of the SO spectrum required ∼170 h of CPU time on a single 2 GHz CPU. For the nonautonomous case, the expression for

*t*) cannot be simplied, and the computations are ∼100 times more expensive than the autonomous case, so will not be considered here.