Predictability & Variability of the Ocean Circulation: A non-normal perspective

Transcription

1 Predictability & Variability of the Ocean Circulation: A non-normal perspective Laure Zanna University of Oxford Eli Tziperman (Harvard University) Patrick Heimbach (MIT) Andrew M. Moore (UC Santa Cruz)

2 Outline Background & Introduction Atlantic Climate Variability & Predictability Linear Algebra Basics, Transient Growth & Optimal Perturbations Optimal excitation of large-scale ocean variability in a GCM: 3D excitation growth ~ 7 yrs Surface excitation growth ~ yrs Conclusions & Implications for Variability and Predictability Zanna, Heimbach, Moore & Tziperman 9a, b (To be Submitted to J. of Climate)

3 North Atlantic SST Variability North Atlantic SST anomaly (smoothed) Atlantic Multidecadal Oscillation = AMO AMO index= 1yr running mean of detrended Atlantic SST anomaly north of the equator (Sutton & Hodson 5; NOAA 9)

4 Meridional Overturning Circulation Estimates of mass 6N (1) 16 ± Sv WOCE (Ganachaud & Wunsch, 1) () 14 ± Sv ECCO (Wunsch & Heimbach, 6) (3) 13 ± Sv GECCO (German ECCO) 195- (Stammer & Kohl, 8) (1), (), (3) used data + simple or complex models Atlantic MOC using the RAPID 6N between 4-6, estimate 18.7 ± 5.6 Sv (Cunningham et al 7) Few direct observations of MOC need to rely on models

5 MOC & SST Variability MOC time series from CCSM3 high lag 5-1yrs (MOC leads the SST) (Latif et al 4) MOC Sea Surface Temperatures (SST) Normalized anomalies Year FIG. 3. Time series of simulated annual mean North Atlantic SST anomalies (4 6 N and Many studies on MOC variability (timescales 3-1 yrs):damped ocn mode excited by stochastic atm forcing, self-sustained oscillations (e.g.,griffies & Tziperman 1995, Marotzke Weaver Sarachik 1991, Chen & Ghil 1995) Variability = damped mode(s) of ocn excited by atm noise

6 MOC Predictability Studies So far MOC predictability: integrations using arbitrary atm. perturbations predictability~yrs (Griffies & Bryan 1997, Latif et al 8) Time [yrs] Instead, which ocn perturbations the MOC is most sensitive to & how uncertainties grow/evolve with time predictability study optimal perturbations & transient growth of MOC (Lohman & Schneider 1999, Zanna & Tziperman 5, Alexander & Monahan 7, Sevellec et al 7, Hawkins & Sutton 9)

7 Stable & P(t) Decay P (t) as t Transient Growth d P (t) dt = A P (t) Unstable P(t) & Growth P (t) as t Time Stable: Growth + Decay Time 3 P(t) Time

8 Which perturbations lead to the fastest growth? Perturbations can grow substantially due to interaction of several non-orthogonal modes, even in absence of unstable modes if system is non-normal Optimal initial conditions = singular vectors fastest growing perturbations leading to an amplification of a given quantity (e.g., Farrell 1988, Trefethen 1991, Buizza & Palmer 1995) Time most fluid dynamical systems are non-normal transient energy growth P(t) Why do we care? Fast growth dp (t) dt Slow decay = A P (t) P (t) as t

9 Linear stable system dp (t) dt Non-normality & Transient Growth = A P (t), P (t) as t If A is non-normal A A T A T A eigenvectors not orthogonal may lead to transient amplification (D) solution at time : P (τ) = a 1 u 1 e λ1τ + a u e λ τ If λ << λ 1 <, then a u e λ quickly leavingmostly P (t> ) a 1 u 1 e λ 1 eventually P ( t ) Transient growth: Interaction of non orthogonal eigenmodes b/c of (1) Partial initial cancellation () Different decay rates slow decay Fast growth P(t) t= t > P(t=)=1 fast decay P(t )>1 Slow decay Time

10 Full nonlinear model d P dt Evaluation of optimal i.c. for MOC = A P, P (τ) = e Aτ P P t = F(P) = B(τ) P Maximize MOC anomalies at t = τ Constraint: I.C. unit norm at t = max P P T B T XB linearized about steady state A= ocn state, P =(T,S,u,v ) P T B(τ) T XB(τ) P P T YP Equivalent to a generalized eigenproblem for optimal i. c. P =1 ( ) { P P 1 } B T (τ)xb(τ) P λ P T Y = λy P T & t = (e.g. Farrell, 1988)

11 Analysis of optimal initial conditions for MOC Spatial structure of the optimal i.c. P with largest eigenvalue λ leading to max growth of MOC for each τ Analysis of optimal perturbations & transient growth i. where & how the MOC is sensitive to perturbations ii. physical mechanism related to ocean/climate variability iii. error growth & predictability limit, can be used for MOC predictions to explore uncertainties Previous Applications: atmospheric cyclogenesis, turbulence, ocean waves & more (e.g., Farrell & coauthors 1988,1989,1993 9), numerical weather prediction (e.g., Palmer & coauthors 1996), ENSO (e.g., Penland & Sardeshmukh 1995, Moore & Kleeman 1997)

12 Optimal Perturbations= ENSO precursor Coupled Model t= Observations (Moore & Kleeman 1997) (Penland & Sardeshmukh, 1995) t= 8 months

13 How to actually do it with a GCM? Solving B( τ) T XB( τ) P = λy P for optimal initial conditions is simple as long as the matrix B (propagator) is known P More complex for large system where evaluation of B is expensive (e.g., GCMs) different ways to approach this problem: o Exact solution of full GCM eigenmodes using tangent linear & adjoint & ARPACK (Golub & Van Loan 1989, Lehoucq et al. 1998): technically challenging & high computational cost o Approximate solution with EOF reduced space (linear inverse modeling (Penland): (1) physics difficult to extract; () EOFs do not form an optimal basis; (3) assumes residuals are white noise

14 Model Configuration Sea Surface Temperatures 6N SST [ C] 5 4N N 15 1 S 4S 5 (NASA model) 6W 45W 3W 15W (MITgcm, Marshall et al. 1997) Solving for the optimals with GCM = using tangent linear & adjoint models (Giering & Kaminski,1998) 6S

15 Model base state for optimal I.C. study Ocean MITgcm: 3 x3,15 levels; mixed b.c.; annual avg 6N (b) Velocity at 17m 1 cm/s x 1 5 6N 1 cm/s (b) Velocity at 495m.5 x 1 4 6N 1 SST [ C] 5 4N 4N 4N N.5 N N 15 [m/s] [m/s] S.5 S S 1 4S 4S 4S 5 6S 6W 45W 3W 15W 1 6S 6W 45W 3W 15W 6W 45W 3W 15W (e.g., Marshall et al. 1997) Model in a stable steady state & any perturbation eventually decays 1 6S

16 MOC Results Overview B T (τ)xb(τ) P = λy P Maximization: (X) sum of squares of MOC anomalies 51N-6N, 17m-35m@ (Bugnion & Hill 6) t = τ Constraint: (Y) unit norm i.c. of T & S weighted by relative volume & contributions to density 1km km 3km 4km 5km Steady state MOC [Sv] Steady State MOC 6S 4S S Eq N 4N 6N 1 1 3D optimal i.c. of T & S: 7.3 yrs (1% mean) MOC as fct of time when initializing with optimals Time [yrs] 3

17 MOC Optimal Initial Perturbations Initial conditions: in deep ocn, NH, with baroclinic structure Both T & S needed for growth & correlated density = independent variable Timeseries of MOC ^ initialized with optimal i.c Year 3 Zonally Zonally averaged averaged density! anomalies anomalies 1km km 3km 4km S Eq N 4N 6N 1! anomalies 3 6N 5N 4N 3N N Density 3m 1N 6W 35W 1W

18 Role of East-West Density MOC proportional to east-west density gradients (at depth) Thermal wind 6N 5N 4N 3N N Density! 3m 1N 6W 35W 1W v z = g ρ fρ x MOC index 3-1!.5 - zonal density 3m! Time [yrs]

19 MOC Maximum Amplification Time We need to explain: 1km km 3km 4km - MOC < - Negative east-west density gradient increases & becomes positive MOC anomalies MOC 7.3 yrs 1 3 Time [yrs] 5km S Eq N 4N 6N 6N 5N 4N 3N N 1N 6WDensity! anomalies 3m 1W 6N 5N 4N 3N N Density anomalies! 3m t= t=7.3 yrs 3 1N 6W 35W 1W

20 MOC Mechanism for Maximum Amplification 65N t= t=7.3 yrs ρ < -m 65N -4m ρ x > v z < ρ < 4N 4N ρ(t = ) < propagates westward & southward Replaces positive density anomaly near western boundary E-W density gradient > MOC anomaly <

21 What causes the westward propagation? Possible answers: Advection by the mean flow? Rossby wave in stratified fluid on beta plane? Linearized equation for density perturbations: ρ t + u ρ x + ρ ρ ( v ρ + u + v y x y ). ( ) Thermal wind: ( u,v ) ( gh ρ gh ρ ) =, f ρ y f ρ x h = anomaly z-scale ρ ( t + u gh ρ f ) ρ ρ ( y x + v + gh ρ f ) ρ ρ x y mean flow & horizontal mean density gradient!

22 Westward propagation due to N-S density velocity of propagation= gradient ( u gh ρ f ) ρ y North ρ + North- South East-West ρ y > ρ < ρ Geostrophy anticyclonic flow around negative density anomaly Southward advection of dense water to the east & Northward advection of light water to the west westward propagation

23 Westward propagation due to N-S density gradient ( velocity of propagation= u gh ρ f ) ρ Horizontal density y gradient acts as background PV gradient y x North ρ + ρ y > ρ < ρ Thermal Rossby wave: related to multidecadal Atlantic variability(colin de Verdiere & Huck 1999, Raa & Dijkstra, Frankcombe 8) non-normality can play a role in controlling the growth of these modes & their potential predictability

24 MOC 3 rd 3 yrs 6N 5N 4N 3N N 1N 6W! anomalies 35W 1W 6N 5N 4N 3N N 1N 6W! anomalies 35W 1W 6N 5N 4N 3N N Density! 3m 3 3 t= months t=7.3 yrs t=3 yrs 1N 6W 35W 1W ρ(t = ) > propagates to western boundary in 3 yrs E-W density gradient< MOC > 1km km 3km 4km MOC 3 yrs MOC anomalies 5km 1 3 S Eq N 4N 6N

25 MOC Heat transport, SST & Energy Growth of meridional Heat Transport transport 55N Heat transport anomaly Time [yrs] high & low latitudes due to the growth of the MOC & T (.1C 8% mean) 1 x Energy Growth = transformation of available potential energy into kinetic energy Time [yrs]

26 MOC Robustness Mean Flow: 1 or overturning cells, symmetric or not 1 st optimal quite similar Maximization: (X) different sum of squares of MOC anomalies similar optimals & mechanism Steady state State MOC MOC Steady Steady state State MOC MOC Steady state State MOC MOC 1km km 3km 4km 1km km 3km 4km 1 1 5km 5km 6S 4S S Eq N 4N 6N Linearity assumption: 6S 4S S Eq N 4N 6N Mechanism works for small perturbations only; Amplitude & sign are arbitrary; 6S 4S S Eq N 4N 6N But similar mechanism observed in nonlinear model for small perturbations (~.1C), amplification slightly reduced

27 How to excite a structure similar to the optimal perturbations?? Mesoscale eddies or deep high latitudes near boundaries Temperature 3km depth (ECCO, MITgcm) (Forget, Heimbach & Wunsch, MIT) but excitation of srfc anomalies by the atm is very likely

28 MOC Surface optimals B T (τ)xb(τ) P = λy P Constraint: (Y) unit norm i.c. of surface T & S only Optimal i.c. of SST & SSS leading to MOC growth ~ 19yrs!!""#!$%!%& "!"""!$%!%& Time evolution of MOC anomalies when model initialized with surface T & S ( +, ( +, initial.1c 19 yrs (9% of the mean) ) * +( +) +* ) * +( +) +*!*!+*!+)!*!+*!+) 1!)!+(!)!+( Time [Yrs]!(!(!)!*!+,!(!(!)!* Exciting only the upper ocean leads to a slower growth of anomalies than deep excitation & to an overestimate of predictability time!+,

29 SST Optimal growth of North Atlantic SST Simple stochastic climate model (Hasselman 1976) T = SST ; 1/λ damping time scale of ocn mixed layer η(t) = white noise forcing Response = no oscillations, red spectrum & predictability = persistence exp( λt) T t = λt + η(t) Ocean is passively responding to the atmosphere

30 SST Optimal growth of North Atlantic SST [m C ] Time evolution of SST^ anomalies when model initialized (b) SST with timeseries optimal SST 3 1 SSTx1.6 in 15 yrs Time [yrs] B T (τ)xb(τ) P = λy P Maximization & Constraint: (X=Y) Northern Hemisphere SST

31 [m 3 Sv [m 3 C ] 4 SST Optimal growth of North Atlantic SST Time evolution of SST^ anomalies when model initialized (b) SST with timeseries optimal SST 3 1 SSTx1.6 in 15 yrs Time [yrs] I.C. = upper ocn temperatures 6N 35N 1N 6N 35N 1N Timeseries of the cost function, (a) J MOC defined by Eq. 1, and (b) J SST defined by hen initializing the linearized model with the surface optimal initial conditions. 6N Growth governed by western boundary vertical stratification & interior N-S temp. gradient Small correlation btw SST (AMO index) & MOC growth 35N 1N 6N 35N (a) α <T> z at t= (b) t=.5 yrs (c) t=8 yrs (d) t=15 yrs 1N 6W 35W 1W x 1 x 1 x 1 3 x 1 x 1 3 x 1 Ocn is dynamically active in amplifying North Atlantic SST FIG. 4. Time evolution of α T z anomalies north of 1N at (a) t =, ( (independently of the MOC growth) t = 8 yrs and (d) t = 15 yrs when the linearized model is initialized with opt maximizing the SST.

32 Optimal Perturbations using LIM GFDL CM.1 HadCM3 Integrated Temperature years (Tziperman, Zanna & Penland 8) (Hawkins & Sutton, 9) Integrated Salinity Integrated Density

33 MOC Optimal growth of MOC anomalies Caveats & Future Research Seasonal Cycle, Dynamical atmosphere, Sea-Ice: can modulate the growth of anomalies; additional feedback Configuration/Resolution: bathymetry source of nonnormality (Chhak et al 6); affect wave properties ; EDDIES? + help to compare optimals & growth mechanism with state estimates/observations Ocean permanently forced: Stochastic optimals sustaining the MOC variability more relevant to climate variability than optimal initial conditions

34 MOC Optimal growth of MOC anomalies Conclusions & Implications Stable system: small initial deep density perturbations LARGE transient growth of MOC anomalies in ~ 7 yrs - Optimal i.c. = deep density anomalies in Northern Hemisphere - Mechanism= propagation mean advection & thermal Rossby waves Non-normal ocean dynamics important on interannual/ multi-decadal climate variability? Growth time scale = predictability limit for MOC ~ 7 yrs Surface perturbations lead to an overestimate of predictability Error at depth in i.c, processes (e.g., eddies overflows) limit predictability Optimal i.c. can be used in GCMs to evaluate predictability similarly to numerical weather predictions