Download Non normal Perturbation Growth of Pure Thermohaline Circulation

Non normal Perturbation Growth of Pure Thermohaline
Circulation using a 2­D Zonally Averaged Model
JULIE ALEXANDER School of Earth and Ocean Sciences
University of Victoria
Victoria, BC, Canada
AND ADAM H MONAHAN
School of Earth and Ocean Sciences
University of Victoria
Victoria, BC, Canada
and
Canadian Institute for Advanced Research
Earth Systems Evolution Program
Submitted to J. Phys. Oceanogr.
Corresponding author address:
School of Earth and Ocean Sciences, University of Victoria,P.O. Box 3055, STN CSC, Victoria BC, Canada V8P 5C2 ([email protected], [email protected])
ABSTRACT
Generalized linear stability theory is used to calculate the optimal initial conditions that result in transient amplification of the thermohaline circulation (THC) in a zonally­averaged single basin ocean model. The eigenmodes of the tangent linear model verify that the system is asymptotically stable but the nonnormality of the system permits the growth of perturbations for a finite period through the interference of nonorthogonal eigenmodes. It is found that the maximum amplification of the THC anomalies occurs after about 35 years with both the thermally driven and salinity driven components playing a major role in the amplification process. The amplification process in this study cannot be attributed to a few eigenmodes of the linear tangent operator but instead include contributions from the interference of almost all of the 800 eigenmodes.
1. Introduction
The thermohaline circulation (THC) is an important feature of the global climate system, transporting a large amount of heat from low latitudes to high latitudes (approximately 1 x 1015 W according to Ganachaud and Wunsch (2000)). Evidence from general circulation model (GCM) simulations (Manabe and Stouffer 1995) and paleoproxies (Clark et al. 2002; O’Hare et al. 2005) suggest that variations in the intensity of the THC are likely to change the climate significantly. The main time scales and mechanisms of variability of the THC are not well known and considerable uncertainty remains in the stability of the present THC. Simulations with climate models can provide useful information regarding the dominant timescales of THC fluctuations, both in terms of internal variability and adjustment to external forcing. In a study by Vellinga and Wood (2002), the HadCM3 ocean­atmosphere general circulation model was used to study climate feedbacks after a major disruption of the THC. In this study the THC was made to collapse by applying strong fresh water forcing to the top layers of the North Atlantic. After about 100 years the model’s THC had largely recovered and most climatic anomalies had disappeared. Using the Geophysical Fluid Dynamics Laboratory climate model Delworth et al. (1993) found the periods of maximum variance of the THC to be between 70 and 100 years. In a further study by Timmermann et al. (1998) it was shown that the dominant variability of the THC had a period of 35 years. This multidecadal variability was identified as the THC responding to changes in upper­
ocean density, caused by near­surface salinity changes over the North Atlantic sinking regions.
In addition, there is evidence gathered from paleoclimate data that the climate system of the Earth has undergone dramatic and abrupt changes in the past (Manabe and Stouffer 1999). High resolution records indicate that large amplitude changes have occurred on millenial timescales, with transitions occurring on timescales of decades (Delworth and Greatbatch 2000). There is a possibility (albeit slight) that future changes of the climate system may exhibit similar nonlinear behavior (Meehl et al., 2007); such changes cannot be easily extrapolated from ongoing observed changes. A characteristic of nonlinear behavior such as the collapse of the global THC is very low predictability. Therefore it is important to increase our understanding of the variability of the THC at points both near and far from nonlinear thresholds. Broecker (1997) has suggested that fluctuations of the thermohaline circulation in the North Atlantic may have induced some of the rapid climate changes during the last ice age. Changes in the thermohaline circulation would seriously affect the climate in eastern North America, western and central Europe, as well as impacting the ocean's biogeochemical cycles and future sea level. Paleoclimate evidence suggests that the THC has been remarkably stable over the past 10,000 years (Tziperman 1997) with present day variability estimated to be between 5% and 10% of the mean transport (Pasquero and Tziperman 2004), suggesting that the present day THC can be characterized as a stable linear dynamic system which may be excited by stochastic atmospheric forcing (Griffies and Tziperman 1995; Pasquero 2004). This statement has the implication that the normal modes of the linearised dynamics, dx / dt = Ax ,must all have exponentially decaying amplitudes so any initial perturbation must eventually decay, where x is the perturbation state vector (anomalies around some steady state solution) and A is the linearised dynamical operator. Traditional linear stability theory focuses on the eigenmodes of the tangent linear operator A and searches for perturbations that grow exponentially and continuously over time (e.g. Pedlosky 1987). If all of the eigenvalues of A are negative, then the system is asymptotically stable and all small perturbations applied to the steady state will decay to zero as t → ∞ . If A possesses at least one growing eigenmode, then the tangent linear dynamical system is considered to be asymptotically unstable. However, this view of stability focuses on the long­term behaviour of the linear system; more complex behaviour is possible over finite times. If A is normal, i.e. it commutes with its adjoint, then its eigenmodes are orthogonal and no perturbation can grow faster than the most unstable eigenmode (Moore et al. 2003). However, if A is non­normal (does not commute with its adjoint), it is possible for perturbations to grow more rapidly than the most unstable eigenmode. In particular, if all of the eigenmodes of a nonnormal system are asymptotically stable it is still possible for the perturbation vector in a non­normal system to undergo a transient growth at any given finite­time before its eventual decay (Farrell and Ioannou 1996). This transient growth is due to the linear interference of the nonorthogonal eigenmodes of the matrix A . In the Euclidean norm the adjoint of a matrix is the Hermitian transpose, but applications often require using a norm other than the Euclidean norm. Farrell and Ioanou (1996) show that by an appropriate transformation of dynamical operators the analysis can proceed as in the Euclidean norm with the implication that it is possible to render an operator normal by choosing an appropriate norm. However this norm is usually not based on physical grounds and it is very unlikely that an operator that is non­normal in the Euclidean norm will become normal under a physically based norm. The linear superposition of eigenmodes that yields the fastest growing perturbation with respect to a given norm and time interval is called an “optimal perturbation” (Farrell and Ioannou 1996). Because optimality is defined relative to a particular norm, the optimal perturbtions are norm­dependent (in contrast to the normal modes). By studying optimal perturbations, disturbances that grow rapidly with respect to a given norm can be identified and draw attention to dynamically interesting parts of the flow. As most geophysical systems are in fact non­normal, the techniques of generalized stability theory have been applied to a variety of problems. It has been shown in numerous studies (e.g. Farrell and Ioannou 1993a, b; Farrell and Ioannou 1995; Farrell and Ioannou 1996; DelSole and Farrell 1995; Aiken et al. 2002; Moore et al. 2003) that a non­normal system responds differently to steady forcing than does a normal system.
A number of studies have investigated the non­normal dynamics of the THC (Lohmann and Schneider 1999; Tziperman and Ioannou 2002; Zanna and Tziperman 2005; Sevellec et al. 2006, manuscript submitted to J. Phys. Oceanogr.). This study investigates the transient amplification of the THC using an ocean circulation model with a more sophisticated representation of the ocean dynamics than those used in previous studies. A hierarchy of climate models has been developed to study the structure and variability of the THC (see the review by Weaver et al. 1999). Although simple box models provide a means of exploring basic physical processes of the THC, (e.g. Stommel (1961), which used such a model to demonstrate that it is possible for the THC to have multiple equilibria for identical surface forcing), they have severe limitations. Considering the ocean as large, well­mixed reservoirs prevents modeling of deep water formation in specific areas and the subsequent upwelling through the stratified thermocline. The model used in this study is the zonally averaged ocean only model of Stocker and Wright (1991), Wright and Stocker (1991) and Wright et al. (1995). Unlike the box models considered in Lohmann and Schneider (1999), Tziperman and Ioannou (2002), and Zanna and Tziperman (2005) this model is on a spherical domain, takes into account the rotation of the earth, and employs a direct representation of the momentum budget. Although Sevellec et al. (2006, manuscript submitted to J. Phys. Oceanogr.) use a similar 2­D zonally averaged model for their study, there are a number of differences between their study and ours. In particular, they consider only salinity perturbations and neglect temperature perturbations. In the configuration considered in this study, this model simulates the THC as driven by buoyancy forcing alone; the effects of surface wind forcing have been neglected. In this sense, this study focuses on the dynamics of pure thermohaline circulation.
The paper is organized as follows. In Section 2 the model is described. The linear theory is presented in Section 3. Section 4 describes the optimal initial conditions and the evolution of the optimal perturbation to the THC. A discussion and comparison to previous studies are given in section 5. Conclusions follow in Section 6.
2. Model description
The ocean model considered in this study is a 2­D zonally averaged model developed by Wright and Stocker (1991) as a model of reduced complexity for the study of the structure and variability of the large­scale THC. This model, which we shall refer to as the WS model, has been shown to be capable of reproducing the major features of the present­day THC (Stocker and Wright 1991; Wright and Stocker 1991; Wright et al. 1995). Numerous studies have used this model to investigate the mechanisms involved in the variability of the THC, the existence of multiple equilibria, and transitions between these equilibria (e.g. Aeberhardt et al. 2000; Stocker et al. 1992a and b; Schmittner and Stocker 2001; Schmittner and Weaver 2001; Knutti and Stocker 2002). The WS model has also been shown to be sensitive to changes in internal parameters in a manner similar to OGCMs (Wright and Stocker 1991; Knutti et al. 2000). The sensitivity of the THC to switches between restoring and mixed boundary conditions and the amounts, rates and locations of freshwater input required to trigger a conveyor shutdown have been extensively investigated using the WS model (Stocker and Wright 1991; Wright and Stocker 1991; Stocker et al. 1992a and b; Vellinga1996) as have the possible impacts of global warming on the atmosphere­
ocean system (Schmittner and Stocker 1999; Stocker et al. 1994; Stocker and Wright 1996). Centennial scale, periodic behaviour of the THC about the steady state was observed by Schmidt and Mysak (1996) when applying linear stability analysis to the WS model. This earlier study focused on the eigenstructure of the linearised dynamics and did not consider transient amplification of THC anomalies.
The state variables of the WS model are temperature and salinity. The model dynamics follow from the Boussinesq equations formulated on a spherical coordinate frame, zonally averaged between the east and west boundaries of the ocean basin of angular width ∆Λ . In the following equations, all state variables are zonally averaged.
The prognostic equations are the conservation of energy and salt expressed as advection­
diffusion equations:
∂T 1 ∂  cυ  ∂
1 ∂  cK H ∂T  ∂ 
∂T 
 2
 +  K V
+
 T  + ( wT ) =

∂t c ∂φ  a  ∂z
c ∂φ  a ∂φ  ∂z 
∂z 
(1)
∂S 1 ∂  cυ  ∂
1 ∂  cK H ∂S  ∂ 
∂S 
 2
 +  K V
+
 S  + ( wS ) =

∂t c ∂φ  a  ∂z
c ∂φ  a ∂φ  ∂z 
∂z 
(2)
Horizontal momentum balance is represented by geostrophic balance without horizontal or vertical diffusion of momentum: 2 sΩu = −
1 ∂p
ρ * a ∂φ
(3)
− 2 sΩυ = −
1 ∆p
ρ * ac ∆Λ
(4)
The assumption of balance does not imply that momentum diffusion is unimportant in determining zonally averaged oceanic conditions. In the present formulation momentum diffusion is implicitly taken into consideration by defining a relationship between the east­west pressure difference and the north­south density gradient as will be discussed below. Vertical momentum balance is represented by hydrostatic equilibrium:
∂p
= − ρg
∂z
(5)
and mass conservation is represented by the continuity equation:
1 ∂
∂
(cυ ) + w = 0
ac ∂φ
∂z
(6)
A linearized equation of state is used to close the system. ρ = ρ * [1 − α ( T − T0 ) + β ( S − S 0 ) ]
(7)
where α = 0.223 kg m −3 K −1 , β = 0.796 kg m −3 ppt −1 and ρ * = 1027 kg / m 3 is a constant reference density. In the above equations, φ is the latitude, s = sin φ , c = cos φ and z is the vertical coordinate, increasing from − H at the bottom to zero at the surface; u ,υ and w are the zonally averaged horizontal and vertical velocity components; T , S ,ρ and p denote zonally averaged potential temperature referenced to the surface, salinity, in situ density and pressure; ρ * = 1027 kg / m 3 is a constant reference density; Ω and a are the angular velocity and radius of the Earth, and g is the acceleration due to gravity. The constant horizontal and vertical diffusion coefficients are respectively K H = 1.0 × 10 3 m 2 s −1 and K V = 0.2 × 10 −4 m 2 s −1 . For this study the model has 20 vertical layers and 20 latitudes in the Northern Hemisphere only. It is a single Boussinesq ocean basin with a uniform depth of 5000m and constant angular width of 60o. It is not possible to determine the east­west zonally averaged pressure gradient, ∆p
, from ∆Λ
the model equations given above. Wright and Stocker (1991) parameterized this pressure gradient in terms of the north­south density gradient. This parameterization guarantees that u , v = 0 at the lateral boundaries and is derived by adding a linear damping term, − µv to Eq. (3) and solving for the pressure gradient. The final expression for the east­west pressure gradient used in this study is, ∆p
∂p
= −ε sin ( 2φ )
∆Λ
∂φ
(8)
where ε is a closure parameter that depends on the width of the basin. Wright and Stocker (1991) provide a discussion of the test of the validity of this parameterization. This closure scheme was developed for models with no wind stress. In order to simplify the analysis, a value of ε = 0.45 , (as used in Wright and Stocker (1991)), was chosen for this study.
No­flux conditions are specified for T and S at solid boundaries: ∂T ∂T ∂S ∂S
=
=
=
=0
∂s ∂z ∂s ∂z
(9)
Furthermore, rigid boundaries are assumed: υ = 0 at northern and southern walls and w = 0 at the top and bottom of the basin. The vertical flux of heat and salt at the surface are parameterised by
− K V Tz =
HM
( T − T* )
τ
(10)
− KV S z =
HM
( S − S* )
τ
(11)
where H M = 50m is the depth of the surface layer and τ = 50 days is the relaxation time scale used for both temperature and salt. Surface temperature and salinity are restored to observed fields of annual mean temperature, T∗ , and salinity, S ∗ , as compiled by Levitus et al. (1994).
Finally, the meridional overturning stream function Ψ is defined by
υ=−
1 ∂Ψ
c ∂z
(12)
w=−
1 ∂Ψ
ac ∂φ
(13)
with Ψ = 0 at the top and bottom of the basin for all latitudes.
The solution procedure is as follows. The temperature and salinity fields are calculated using forward time­differencing and numerical diffusion associated with the advective terms is reduced using the method described in Wright and Stocker (1991). The density field is determined by the linear equation of state and from this the streamfunction and velocity fields can be calculated as described in Wright and Stocker (1991).
Static instabilities are removed using the convection scheme described in Wright and Stocker (1991). This is an efficient mixing scheme that parameterizes small scale motions not resolved by the model. Unstable stratification is removed by determining the top and bottom cells of unstable parts of the water column and setting the temperature and salinity to the volume­
weighted mean over this region. The effect of convection is to reduce vertical density gradients and mix down any positive density anomalies on very short time scales. For all runs of the full nonlinear model the vertical water columns were swept four times to remove static instabilities. 3. Linear theory
In this paper the techniques of generalized linear stability theory (Farrell and Ioannou 1996) are used to calculate the linear perturbations that lead to the most rapid growth of THC anomalies. The system of governing equations may be written in terms of linear and nonlinear operators L and N respectively
N
dP(t )
= N(P(t )) + L(P(t ))
dt
(14)
where P is the state vector of the system consisting of the temperature and salinity at each grid point. If a perturbation P′ is added to the steady state solution P then the perturbation evolution is described by dP ′  dN 
=
 P′ + N(P′) + L(P′) = AP′ + N(P′) ≈ AP′
dt  dP  P
(15)
assuming that P′ is sufficiently small so that N(P′) is negligible compared to AP′ .
The tangent linear operator A which represents the linearized model equations is defined as
 dN 
A =L+

 dP  P
(16)
A time­independent steady state is used to simplify the analysis and dynamical interpretation of the system. In this case the system is autonomous and its eigenmodes evolve exponentially in time. The solution of the tangent linear system is P′(t ) = e AtP′(t = 0) = B (t ,0)P′(0)
(17)
where the matrix B (t ,0) = e At ≡ B (t ) is known as the propagator of the system since it advances the state vector forward in time and P′(0) ≡ P0′ is the perturbation state vector at t=0.
4. Optimal initial conditions
The full nonlinear model is started from rest with the ocean at a uniform temperature of 4oC and salinity of 35ppt and run to steady state for 12,000 years under the restoring boundary conditions described in Eq. (12) and Eq. (13). Figure 1 shows the steady state streamfunction, temperature and salinity fields reached by the full nonlinear model. The streamfunction exhibits a typical overturning circulation with a maximum value of only 4.3 Sv. Although this is considerably lower than the observed 15Sv it is consistent with results obtained using this model in the absence of wind forcing (Stocker et al. 1994). It was found in Stocker et al. (1994) that the overturning strength in the Atlantic almost doubles when the winds are turned on with all other model parameters remaining constant. The linear tangent matrix is calculated using Eq. (16) and a perturbation of 0.0001 oC and 0.0001 ppt in the prognostic equations. The eigenvalues of A are all negative indicating a stable system: any initial perturbation to the system will decay to zero after a sufficiently long time, with a decay timescale for each mode given by the reciprocal of the real part of its eigenvalue. Modes with imaginary eigenvalues will oscillate as they decay with timescales of 2PI times the reciprocal of the imaginary part of the eigenvalue. Figure 2 shows the decay and oscillatory timescales of the 800 modes of the system. The decay timescales range from days to thousands of years. Of the 800 modes, 492 oscillate as they decay with oscillation timescales ranging from decades to thousands of years. The decay timescale appears to be independent of whether or not the mode oscillates. This differs from the results obtained from Zanna and Tzipermann’s (2005) coupled atmosphere­ocean model, in which it was found that the very fast decaying modes did not oscillate.
As discussed in Tziperman and Ioannou (2002) a quantity representing the THC amplitude must be defined in order to study the growth of anomalies by the nonnormal linearized dynamics in the stable regime. This quantity, U (φ ) , can be written in terms of the depth averaged meridional velocity, v (φ ) U (φ ) = v(φ )∆xH top
(18)
In the above, ∆x is the width of the basin and
v(φ ) =
1
H top
0
∫ v(φ , z )dz
(19)
− H top
where H top = 1800m . A plot of U (φ ) as a function of latitude for the steady state solution is shown in Figure 1. The maximum value occurs at about 72o N which is consistent with the steady state streamfunction illustrated in Figure 1.
The quantity to be maximized, N (t = τ ) , is the sum over latitudes of the squares of the THC anomaly amplitude, 72o
N (t = τ ) = ∑ U (φ , t = τ )
2
(20)
φ =0
(from this point onwards, all state variables are perturbation quantities so primes are dropped for notational convenience). Evaluating the THC anomaly at the single latitude (72oN) where U (φ ) is a maximum rather than integrating over all latitudes did not yield particularly interesting results: in this case transient amplification and then decay did not occur. Instead, the THC anomaly was initially nonzero and simply decayed over time such that large anomalies occurred only at 72oN rather than over the whole domain. An analogous result was obtained in Zanna and Tziperman (2005). The upper end of integration in the definition of the THC anomaly norm ends at 72oN in order to exclude high latitudes with weak stratification where convective overturning occurs. The parameterised convection involves a stepfunction nonlinearity that cannot be linearised. By focusing the measure of tangent linear model norm on latitudes where convection does not occur (for sufficiently small perturbations), difficulties in accounting for convective adjustment are avoided.
In the linearized dynamics the THC amplitude is directly proportional to the statefunction according to U (φ , t ) = Rφ P(t ) . The norm used to measure the growth of the state vector anomaly at time τ , P(τ ) , follows from Eqn (20) with N (τ ) = P T (τ ) XP(τ ) where the norm kernel matrix X is
72 o
X = ∑ ( Rφ RφT )
(21)
φ =0
A measure of perturbation growth or decay is the factor by which the norm changes over the time interval τ , and is given by N (τ ) P T 0 B T (τ ) XB(τ )P0
λ=
=
N (0)
P T 0 XP0
(22)
where B (τ ) is the propagator of the tangent linear equation, such that P(τ ) = B (τ )P0 . The optimal perturbation is the perturbation that yields the largest value of λ subject to the constraint
N (0) = 1 . According to the Rayleigh–Ritz method, the optimal perturbation is the eigenmode of B T (τ ) XB(τ ) with largest eigenvalue (Moore et al. 2003). The initial conditions P0 leading to an optimal growth at time τ are therefore the generalized eigenvectors of the generalized eigenvalue problem
T
B (τ ) T XB(τ )e = λXe subject to P0 XP0 = 1 (23)
Because temperature and salinity perturbations can counteract each other so as to have no net effect on the strength of the overturning, Eq. (20) results in a singular matrix (i.e. its determinant is zero) that in turn may result in infinite amplification factors over finite time. In a transient growth analysis done by one of the authors for the THC in Griffies and Tziperman’s (1995) 4 box model the singular norm kernel defined in Eq. (20) leads to 2 of the 8 eigenmodes having infinite amplification. This nonuniqueness of the optimal eigenmodes may result in a suboptimal transient growth estimate (Tziperman and Ioannou 2002). To eliminate this possibility Tziperman and Ioannou (2002) regularize the norm kernel by adding a small diagonal matrix to X to create a nonsingular matrix. This technique was also applied to X in this study resulting in a matrix with small but nonzero determinant. The timescale for optimal growth, τ opt , is the value of τ maximizing the amplification factor λ ; a plot of λ as a function of τ is given in Figure 3. The maximum value for the largest eigenvalue occurs at τ opt = 35 years. This is the timescale used in the propagator matrix for all subsequent calculations. The optimal salinity and temperature perturbations calculated from the linearized model equations that maximize the transient amplification of the THC anomaly for tau = 35 years are shown in Figure 4. The following discussion will focus on perturbations which amplify the THC strength; perturbations of opposite sign to these will attenuate the THC strength in a symmetric manner. Note that because the dynamics are linear, the overall amplitude of the perturbation is of no intrinsic significance. The initial perturbations consist of cold salty water in the top 1000m of the ocean with warm fresh water in the deep ocean, such that the largest anomalies occur between the latitudes of 60oN and 85oN. This positive temperature anomaly then amplifies for about 35 years before decaying. The positive salinity anomaly at the surface exhibits maximum growth at the very high latitudes with the growth occurring at mid depth for slightly longer than the growth of the temperature anomaly. The negative salinity anomaly grows at low latitudes in the deep ocean and does not decay as rapidly as the high latitude positive salinity anomalies.
The evolution of the THC anomaly, U (φ , t ) = Rφ P(t) , and the contributions to the THC anomaly from the salinity and thermally driven components are displayed as a function of latitude in Figure 5. Initially the salinity and thermally driven components exactly cancel each other. As the perturbation evolves, there is a growth of partially compensating temperature and salinity anomalies such that the most significant THC anomaly growth occurs at 72oN. Ultimately, the salinity component dominates the growth of the THC anomaly. Throughout most of the evolution of the perturbation, anomalies in each of temperature and salinity take opposite signs in middle and high latitudes between 60oN and 75oN. Initially the salinity component of the THC is negative while the thermal component is positive. At low latitudes these components change sign after about 100 years (Figure 6). At higher latitudes this crossover occurs sooner. The participation of both the thermally driven and salinity driven component of the THC anomalies is significant. As seen in Figure 7, the contribution from the salinity component alone produces double the amplification of the THC as that which results from the contribution of both the thermal and salinity components (Figure 8). The growth of the salinity and thermally driven components of the THC tends to occur for a longer period of time than the growth of the total THC, because the two contributing components tend to exactly cancel each other after about 50 years. For the above analyses it has been assumed that the evolution of the THC anomaly is governed by linear dynamics. It is therefore important to compare the evolution of the THC anomaly amplitude, U (φ ) , calculated using the full nonlinear model to that calculated using the linear model. This was done by adding the optimal initial conditions state vector (of small norm) to the steady state solution of the Stocker and Wright model and then running the full nonlinear model for a further 200 years. The THC anomaly amplitude calculated from the nonlinear model exhibits a similar transient growth pattern as that of the linear model (Figure 8). The differences are small enough that we may conclude that the linear approximation is valid for sufficiently small perturbations.
Even though all of the eigenmodes of the linear operator A are stable, the non­normality of A results in nonorthogonal, linearly dependent eigenmodes. Transient perturbation growth results from the linear interference of several if not all of these eigenmodes. The amount of perturbation growth possible is proportional to the degree of linear dependence of the modes (Aiken et al. 2002). To investigate the “optimal” eigenmodes of the linear operator which participate in the observed perturbation growth the optimal initial conditions Po are projected onto the eigenmodes of A (Farrell and Ioannou 1996). The eigenvalue spectra of A and A T are identical each having eigenvalues that occur in complex conjugate pairs. Let { λ n , ŝ n } and {σ n , r̂n } be the {eigenvalue, eigenvector } sets of A and *
A T respectively, where λ n = σ n , and * denotes complex conjugate. The bi­orthogonality property between the eigenmodes of A and A T can be stated as
sˆnT rˆm (λn − σ m* ) = sˆnT rˆm (λ n − λ m ) = 0
(24)
so λ n = λ m for n = m and sˆnT rˆm = 0 for n ≠ m . The state vector Po can be written as a linear superposition of the eigenmodes with projection coefficients a k . Po =
k = 800
∑ a sˆ
k =1
k k
(25)
Since the perturbation has unit norm, P0T XP0 = 1. the amplitude of the k th eigenmode is a k = rˆkT P0 / rˆkT sˆk
(26)
Perturbations with large projection coefficients are the major contributors to the optimal initial conditions. Figure 9 shows the THC amplification as a function of the cumulative sum of modes contributing to the amplification process (computed by finding the norm of the projection of the state vector at the time of maximum amplifiaction on the space spanned by the relevant eigenvectors). Projection coefficients are ordered according to their absolute value. If a k is complex, the conjugate pair is counted as one mode in Figure 9. .It is not possible to reproduce the transient amplification of the THC using only a small number of modes. The maximum amplification factor λ from the contribution of the top 700 modes is only 7.8 compared to 49 for λ
the full set of 800 eigenmodes. The amplification due to the bottom 100 modes alone is only 2.9x10­4, however when these modes are included with the top 700 they increase the amplification from 7.8 to 49. This suggests that there is strong interference between all of the eigenmodes of the linear tangent operator leading to the transient amplification. This result is consistent with the nonnormal dynamics of the system where it is expected that the transient growth is due to the interference of linearly dependent eigenmodes. 5. Discussion
Although the present study is the first to consider optimal perturbation growth of the THC involving both temperature and salinity fields in a model as complex as the WS model, previous studies have considered transient growth in more idealised systems. The first study to apply the ideas of non­normal dynamics to the THC was that of Lohmann and Schneider (1999), in which the nonnormal properties of the Stommel 2 box model of the THC are considered. They found that the optimal perturbation affecting long­term climate variability has little resemblance to the most unstable mode of the system and the leading Empirical Orthogonal Function (EOF). Their results suggest that climatic states which are associated with North Atlantic deep water (NADW) formation are located in a specific part of the phase space only where the NADW circulation with a large meridional salinity or small temperature gradient is not stable. This is supported by Tziperman (1997) who finds inherently unstable climate behavior due to a weak THC. In agreement with the results of Lohmann & Schneider (1999), the present study finds that non­
normal dynamics lead to perturbations involving both temperature and salinity which display pronounced transient growth. The amplification timescale found in Lohmann and Schneider (1999) is shorter than that found in the present study, but detailed quantitative agreement cannot be expected because of the highly idealised nature of the model used in the earlier study.
Tziperman and Ioannou (2002) presented a detailed analysis of non­normal dynamics of the THC using a 3 box model, for which two physical mechanisms for the transient amplification of the THC are identified. One, with a transient amplification time scale of approximately two years, involves an interaction between the THC anomaly induced by rapidly decaying sea surface temperature anomalies and the THC anomaly induced by the slower decaying salinity modes. The second mechanism of transient amplification is primarily driven by an interaction between different slowly decaying salinity modes, and has a typical growth time scale of few decades. For both mechanisms, the initial THC anomaly was zero because of the offsetting effects of temperature and salinity anomalies on the meridional density gradient. While this earlier study agrees with the present study in finding substantial transient amplification on decadal timescales, it differs in that this optimal perturbation is dominated by salinity dynamics.
Transient amplification of the THC in a less idealised model was studied by Zanna and Tziperman (2005),who employed a two dimensional coupled box atmosphere­ocean model to investigate possible physical mechanisms for the transient growth of THC anomalies. This study found transient growth displaying maximum amplification after 42 years with optimal perturbations involving both temperature and salinity anomalies. This more recent study found that the amplification mechanism was the result of an interaction between a few damped (oscillatory and nonoscillatory) modes with decay times between 20 and 800 years. In agreement with this earlier study, the present study finds the optimal perturbation leading to the greatest transient growth involves interactions between temperature and salinity anomalies which combine in the initial conditions to produce an initial THC perturbation of zero. On the other hand, while the present study finds that the optimal perturbation involves significant contributions from a large number of normal modes, Zanna and Tzipermann (2005) find that the transient amplification of the THC in their model is mainly due to the interaction of five slow­decaying modes with timescales of between 23 and 784 years. However, they do find that in order to see the dramatic amplification process many more than the five dominant modes are needed. In a recent study, Sevellec et al. (2006, manuscript submitted to J. Phys. Oceanogr.) use a 2­D latitude­depth ocean thermohaline circulation model to study the variability of the ocean circulation perturbed by a freshwater flux and also find that the nonnormality of the dynamics leads to transient growth of the overturning circulation intensity. However, in the study of Sevellec et al. (2006), only salinity optimal perturbations are used to excite the THC: the temperature component is not included. This study finds that the maximum response of the circulation intensity occurred after 67 years, associated with a single weakly damped linear eigenmode with an oscillation timescale of 150 years. In the present study, with both active temperature and salinity contributions to the density of seawater, the timescales of optimal THC amplification and decay are considerably shorter. Perturbations in temperature and salinity persist for many decades, but in such a way that they counterbalance to produce little effect on the overturning strength after about 100 years. Sevelle et al. (2006) also obtain very different results from truncating the contributions from the eigenmodes of the linear operator, concluding that only a couple of eigenmodes and their biorthogonals determine the effect of initial perturbations on long time scales and that the finite growth for the overturning intensity is due to a single oscillatory mode. The inclusion of a single active component in the density is not the only difference between the model employed in the present study and that of Sevellec et al. The earlier study employs a Cartesian coordinate system, while the present study is in a spherical coordinate system. While the models in both studies are restricted to that of a single hemisphere, Sevellec et al place their northern boundary at 60oN. We find that much of the interesting dynamics occurs at latitudes between 60oN and 75oN. Furthermore, in order to obtain a realisitic steady state overturning circulation Sevellec et al. (2006) have introduced a stong frictional term to the meridional momentum balance. The steady state found in our study is considerably smaller because of the absence of this term, consisent with the meridional transport associated with purely thermohaline circulation in the absence of wind forcing (Stocker et al. 1994). Finally, convection is not explicitly represented in the study of Sevellec et al. (2006).
6. Conclusion
Characterising the variability of the thermohaline circulation is important for understanding internal variability in the Earth's climate system and for detecting trends of anthropogenic origin. In this study we have applied the ideas of generalized linear stability theory to the dynamics of the thermohaline circulation described by a 2­D zonally averaged ocean model. As a result of non­
normality, transient growth of THC anomalies was shown to be possible in an asymptotically stable system. We examined the optimal initial conditions that lead to this transient growth using the tangent linear equation linearized about the steady state, and found that the maximum amplification of the THC anomaly occurs after about 35 years. These anomalies have largely disappeared after about 200 years. The temperature and salinity anomalies evolve over longer timescales than the THC anomaly with the majority of the growth occurring in the high latitude surface ocean. Both the thermally driven and salinity driven components of the THC anomaly play major roles in the amplification process. Initially their contributions tend to cancel each other but when the amplification is at its maximum the salinity driven component dominates producing a negative anomaly at low latitudes and a larger positive anomaly at latitudes between 60 oN and 75oN. After approximately 200 years, the salinity and temperature anomalies again counteract each other and have no net effect on the overturning strength. These results are broadly in agreement with the study of Zanna and Tziperman (2005), who find maximum THC amplification on decadal timescales with contributions coming from both temperature and salinity anomalies.
The amplification process found in this study can not be contributed to a few eigenmodes of the linear tangent operator but instead must be contributed to the interference of a large number of the eigenmodes. This result differs from the results of Zanna and Tziperman (2005) where it was found that the transient growth mechanism was activated by the interaction of only five slow­
decaying eigenmodes and also from the results of Sevellec et al. (2006, manuscript submitted to J. Phys. Oceanogr.) where it was found that a singly weakly damped 150 yr period linear mode explained the major part of the circulation intensity response. The dynamics considered in this study were of pure thermohaline circulation, as wind forcing was not accounted for. In reality, the meridional overturning circulation is strongly affected by surface wind forcing; a natural extension of the present study is to consider optimal perturbation growth in the presence of surface winds.
Another important direction of research is the determination of the relevance of the optimal perturbations to the response to fluctuating surface forcing. This would involve calculation of the stochastic optimals, defined as the spatial structures of the noise that result in maximum variance of the THC variability. Future studies will consider the structure of these stochastic optimals and their projection on the optimal perturbations characterised in this study.
Acknowledgements
The authors would like to thank Laure Zanna for her helpful advice on this research. Julie Alexander would like to acknowledge support from the Canadian Foundation for Climate and Atmospheric Sciences. Adam Monahan would like to acknowledge support from the Natural Sciences and Engineering Research Council of Canada, from the Canadian Foundation for Climate and Atmospheric Sciences, and from the Canadian Institute for Advanced Research Earth System Evolution Program.
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Figure 1 Figure 1. The steady­state solution reached by the full nonlinear model 12, 000 years after spinup. (a) salinity, (b) temperature, (c) meridional streamfunction and (d) linearized THC amplitude Figure 2 Figure 2. (a) Decay time scale and (b) oscillation time scale for the eigenmodes of the linear tangent matrix A as a function of mode number.
Figure 3 Figure 3. Maximum amplification factor (Eqn. 22) of the linearized model as a function of time.
Figure 4 Figure 4. (a) Optimal initial temperature perturbation (T) and (b) optimal initial salinity perturbation (S) calcula ted from the linearized model equations that maximize the transient amplification of the THC for = 35 years. (c) to (h) temperature and salinity perturbations at 5 yr, 35 yr and 100 yrs.
Figure 5 Figure 5. Evolution of the linear THC anomaly (solid line) and its contributions from the temperature (dashed line) and salinity (dotted line) as a function of latitude calculated from the linearized model equations starting from the optimal initial conditions that maximize the transient amplification of the THC anomaly for = 35 years. (a) t=0, (b) t=5 years, (c) t=20 years, (d) t=35 years, (e) t=50 years and (f) t=200 years.
Figure 6
Figure 6. Evolution of the linear THC anomaly (solid line) and its contributions from the temperature (dashed line) and salinity (dotted line) as a function of time calculated from the linearized model equations starting from the optimal initial conditions that maximize the transient amplification of the THC anomaly for = 35 years. (a) =18o, (b) =36o, (c) =54o and (d) =63o.
Figure 7
Figure 7. Evolution of the a) salinity component of the THC amplitude and b) temperature component of the THC amplitude as a function of time and latitude calculated from the linearized model equations starting from the optimal initial conditions that maximize the transient amplification of the THC anomaly for =35 years.
Figure 8
Figure 8. (a) THC anomaly calculated from the linearized model equations and (b) THC anomaly calculated from the full nonlinear model starting from the optimal initial conditions that maximize the transient amplification of the THC anomaly for =35 years.
Figure 9
Figure 10. THC Amplification as a function of the cumulative sum of the top modes participating in the amplification process. (A complex conjugate pair is counted as one mode here).