Nonlin. Processes Geophys., 19, 529–539, 2012 www.nonlin-processes-geophys.net/19/529/2012/ doi:10.5194/npg-19-529-2012 © Author(s) 2012. CC Attribution 3.0 License.
Nonlinear Processes in Geophysics
Predictability of extreme values in geophysical models A. E. Sterk1 , M. P. Holland1 , P. Rabassa2 , H. W. Broer2 , and R. Vitolo1 1 College
of Engineering, Mathematics and Physical Sciences, Harrison Building, Streatham Campus, University of Exeter, North Park Road, Exeter, EX4 4QF, UK 2 Johann Bernoulli Institute for Mathematics and Computer Science, University of Groningen, P.O. Box 407, 9700 AK Groningen, The Netherlands Correspondence to: A. E. Sterk (
[email protected]) Received: 10 June 2012 – Accepted: 27 August 2012 – Published: 17 September 2012
Abstract. Extreme value theory in deterministic systems is concerned with unlikely large (or small) values of an observable evaluated along evolutions of the system. In this paper we study the finite-time predictability of extreme values, such as convection, energy, and wind speeds, in three geophysical models. We study whether finite-time Lyapunov exponents are larger or smaller for initial conditions leading to extremes. General statements on whether extreme values are better or less predictable are not possible: the predictability of extreme values depends on the observable, the attractor of the system, and the prediction lead time.
1
Introduction
Extreme value theory (EVT) for time series {Xi }∞ i=1 studies the limiting distribution of the partial maxima max(X1 , . . . , Xn ) as n → ∞. EVT was originally developed for time series of near-independent random variables (Leadbetter et al., 1983; Coles, 2001; Beirlant et al., 2004), but in the last decade EVT has been extended to chaotic deterministic dynamical systems (Haiman, 2003; Freitas and Freitas, 2008; Freitas et al., 2010; Gupta, 2010; Holland et al., 2012a,b; Felici et al., 2007a,b). In the latter context one considers time series generated by evaluating a scalar observable along evolutions of the system. In this paper we study the predictability of extreme values, rather than their probability distribution. Lyapunov exponents measure predictability, but they are not very useful in forecasting applications. Firstly, Oseledec (1968) proved that Lyapunov exponents are almost everywhere constant functions of the initial condition. This means that all forecast outcomes are equally (un)predictable regardless of the current
state of the system. Secondly, forecasts are typically made ahead for only a few days up to weeks, whereas Lyapunov exponents are asymptotic quantities, which are computed for time tending toward infinity. Finite-time Lyapunov exponents (FTLEs) measure the exponential growth rate of nearby trajectories over a finite time, and typically they strongly depend on the initial condition. Nese (1989) and Abarbanel et al. (1991) showed that FTLEs for dynamical systems such as the Lorenz-63 model and the H´enon map can be very different along various parts of the attractor. Prasad and Ramaswamy (1999) and Datta and Ramaswamy (2003) demonstrated that distributions of FTLEs in intermittent dynamical systems are non-Gaussian, asymmetric, and often have fat tails. Similarly, Lai (2007) found universal distributions for FTLEs in systems having an attractor consisting of two distinct components. For studies on how FTLEs converge to their infinite-time counterparts see Bailey et al. (1997), Ziehmann et al. (1999, 2000), and references therein. In this paper we study whether initial conditions leading to extreme values have larger or smaller FTLEs and how this depends on the prediction lead time and the attractor of the system. Section 2 introduces our concept of FTLEs based on norms that are tailored to the observable of interest, such as energy, convection, or wind speed. In Sect. 3 we study the distribution of FTLEs as a function of lead time and threshold for three geophysical models: a spectral truncation of the barotropic vorticity equation and the Lorenz-63 and Lorenz96 models. Section 4 concludes the paper with a discussion.
Published by Copernicus Publications on behalf of the European Geosciences Union & the American Geophysical Union.
Barotropic vorticity equation Lorenz–96
530
φsid φtop φval φen
235.36 0.00 2.13 399.40
659.17 43.28 430.42 674.20
960.70 1044.14 974.70 1009.50
879.78 191.44 711.20 795.50
891.16 254.23 757.27 814.20
928.99 406.08 851.57 849.50
A. E. Sterk et al.: Predictability of extreme values in geophysical models
Table 1: Descriptive statistics for the observables used with the three models.
0%
95%
97%
99%
12 8 Λτ,q
4 0 -4 -8 0.25
0.5
0.75
1
1.25
1.5
1.75
τ
Fig. 1: Distributions of the set Λ
for the Lorenz–63 model and observable φ
: box plots as a function of lead time τ
τ,q Fig. 1. Distributions of the set 3τ,q q.for the Lorenz-63 model and observable φcon : con box plots as a function of lead time τ for different for different thresholds Boxes indicate the interquartile range, whiskers extend to the minimum and maximum values, and thresholds q. Boxes indicate the interquartile range, whiskers extend to the minimum and maximum values, circles mark the median. For circles mark the median. For each lead time four thresholds are chosen: q = −∞ (gray) and the 95th, 97th, and and 99th percentiles each lead time, four thresholds chosen: q = and −∞ (grey) and the 95th, 97th, and 99th percentiles of the observable (red, green, and blue, of the observableare (resp. red, green, blue). respectively).
2.1 Extreme events
2
Methodology The setting is a system of ordinary differential equations n
x˙ = f (x), x ∈ R , In summary, our methodology consists of checking whether(1) initial conditionsandleading totheextreme have we denote evolution values operator typically by Φt . In addition, consider a scalar observable φ : Rn → R. We define the larger or smallerwefinite-time Lyapunov exponents.
2.1
extreme value domain as the set of points in phase space for which the observable exceeds a threshold q:
Extreme events
Eq = {x ∈ Rn : φ(x) > q}.
The setting is a system of ordinary differential equations
We speak of an extreme event whenever a trajectory Φt (x)
x˙ = f (x),
x ∈enters Rn , the set Eq .
(1)
The papers on extreme value theory for dynamical systems citedevolution in the introduction focused on the case where φ has the and we denote the operator by 8 t . In addition, we form n consider a scalar observable φ φ(x) : R =→ R. We g(dist(x,e x)),define the ex-
treme value domain as the set of points in phase space for where g : [0,∞) → R is a continuous function and x e is a which the observable a threshold q:pointed out by Holland point onexceeds the attractor. However, as et al. (2012b), physically relevant observations are not of this
Eq = {x ∈ Rn : φ(x) > q}.
We speak of an extreme event whenever a trajectory 8t (x) enters the set Eq . The papers on extreme value theory for dynamical systems cited in the introduction focused on the case where φ has the form
form. For example, geophysical quantities such as vorticity or components of the velocity vector can often be written as where Q = Q> is a nonnegative definite n × n
matrix. In φ(x) = q ⊤ x (2) this paper we exclusively focus on observations of the form where(2) q ∈ and Rn is(3). a nonzero vector. Positive quantities, such Eqs. as energy, enstrophy, or squared wind speeds, can often be written as 2.2 Finite-time Lyapunov exponents φ(x) = x⊤ Qx (3) ⊤
where Q = Q is a nonnegative definite n× n matrix. In this Infinitesimal errors in the initial condition x of the system paper we exclusively focus on observations of the form (2) in the direction v evolve as Lt (x)v, where the matrix Eq. (1) and (3). Lt (x) is the solution of the matrix intial value problem 2.2 Finite-time Lyapunov exponents
˙ = Df (8errors X X(0) = I, x of the system (1) Infinitesimal in the initial condition t (x))X,
(4)
in the direction v evolve as Lt (x)v, where the matrix Lt (x) is the solution of thereferred matrix intial problem which is often tovalue as the tangent linear
equation or first variational equation. singular vectors X˙ = Df (Φt (x))X,The X(0) = I, (4) of Lt (x) give the direction and magnitude of maximal error growth along which is often referred to as the tangent linear equation or afirst finite-time segmentTheofsingular an orbit in of state variational equation. vectors Lt (x)space give (Buizza and Palmer, 1995; Palmer et al., 1998; Leutbecher and Palmer, 2008). To measure the growth of perturbations over a time interval [0, τ ], we use different norms at t = 0 and t = τ . Let C be the initial error covariance matrix, and let P be an orthogonal projection matrix (i.e. P = P 2 = P > ). Let k · k denote the Euclidean norm. The Rayleigh–Ritz quotient
φ(x) = g(dist(x,e x )), where g : [0, ∞) → R is a continuous function and e x is a point on the attractor. However, as pointed out by Holland et al. (2012b), physically relevant observations are not of this form. For example, geophysical quantities such as vorticity or components of the velocity vector can often be written as φ(x) = q > x,
where q is a nonzero vector. Positive quantities, such as energy, enstrophy, or squared wind speeds can often be written as
Nonlin. Processes Geophys., 19, 529–539, 2012
(3)
(5)
is maximal if and only if v is the right singular vector of P Lτ (x)C 1/2 associated with the largest singular value. We define the time-τ finite-time Lyapunov exponents (FTLE) as
(2)
∈ Rn
φ(x) = x > Qx,
kP Lτ (x)vk2 v > Lτ (x)> P Lτ (x)v = kC −1/2 vk2 v > C −1 v
λi (x, τ ) =
1 log σi (x, τ ), τ
i = 1, . . . , n,
where σ1 (x, τ ) ≥ σ2 (x, τ ) ≥ · · · ≥ σn (x, τ ) are the singular values of P Lτ (x)C 1/2 . In the following we assume that the initial error covariance C is the identity matrix. www.nonlin-processes-geophys.net/19/529/2012/
A. E. Sterk et al.: Predictability of extreme values in geophysical models
531
Table 1. Descriptive statistics for the observables used with the three models. Model
Observable
Min
Mean
Max
95th percentile
97th percentile
99th percentile
Lorenz-63
φcon φen
−19.13 9.57
0.04 772.42
19.09 2630.22
13.34 1744.66
14.27 1859.84
15.66 2058.29
Barotropic vorticity equation
φsid φtop φval
235.36 0.00 2.13
659.17 43.28 430.42
960.70 1044.14 974.70
879.78 191.44 711.20
891.16 254.23 757.27
928.99 406.08 851.57
Lorenz-96
φen
399.40
674.20
1009.50
795.50
814.20
849.50
We choose the projection matrix P such that the singular vectors will point in the direction of maximal growth of the observable φ. For an observable of the form Eq. (2), we set
3τ,q = { λ1 (x, τ ) | x ∈ S and 8τ (x) ∈ Eq }
qq > P= kqk2 so that kP xk2 =
To study the predictability of extreme events, we define the set
φ(x)2 . kqk2
In case of observable Eq. (3), recall that a symmetric nonnegative definite n × n matrix has k ≤ n positive eigenvalues µ1 ≥ · · · ≥ µk > 0 and the corresponding eigenvectors qi are orthonormal. We can rewrite Eq. (3) as φ(x) = x > Qx =
k X
µi (qi> x)2 .
containing the FTLEs for initial conditions such that after τ units of time the observable exceeds the threshold q. We study how the distribution of this set changes as the lead time τ and threshold q increase. For fixed lead times we compare the distributions of the set 3τ,q for q = −∞ (the entire sample) and for q being the 95th, 97th, and 99th percentiles of the sample { φ(x) | x ∈ S }. If FTLEs cluster to larger positive (smaller negative) values when q increases, then extremes are less (better) predictable.
i=1
In this case we set P=
k X
qi qi>
i=1
so that kP xk2 =
k X (qi> x)2 , i=1
and
3
Results
We apply the methodology of Sect. 2 to three geophysical models: the Lorenz-63 and Lorenz-96 models and a spectral truncation of the barotropic vorticity equation. As observations we will take energy, convection, and squared wind speeds. 3.1
The Lorenz-63 model
µk kP xk2 ≤ φ(x) ≤ µ1 kP xk2 .
In this section we consider the classical model for Rayleigh– B´enard convection derived by Lorenz (1963):
Hence, the φ(x) is large if and only if kP xk is large.
x˙ = σ (y − x),
2.3
with the classical parameter values σ = 10, ρ = 28, and β = 8/3. We use the observables
Predictability of extreme values
We fix an initial point x0 on the attractor and a sampling frequency ω > 0 and compute a sample of points along an orbit on the attractor: S = { 8k/ω (x0 ) | k = 1, . . . , N }. We choose ω and N sufficiently large to provide a good sampling of the attractor as far as both local and global fluctuations are concerned. Typically, we take N = 106 , whereas ω depends on the time scale of the system of interest. www.nonlin-processes-geophys.net/19/529/2012/
φcon (x, y, z) = x
y˙ = x(ρ − z) − y,
z˙ = xy − βz,
(6)
and φen (x, y, z) = x 2 + y 2 + z2
that respectively measure the intensity of convection and the total energy. For the projection matrix in Eq. (5), we respectively take 100 100 Pcon = 0 0 0 and Pen = 0 1 0 . 000 001 Nonlin. Processes Geophys., 19, 529–539, 2012
532
A. E. Sterk et al.: Predictability of extreme values in geophysical models
A.E. Sterk et al.: Predictability of extreme values in geophysical models
ges as the lead time times we compare ∞ (the entire sam99th percentiles of
8
12
6
8
2
6 z
z
0
πb
10
4
er negative) values better) predictable.
4
-2
0
2
-4
0
πb
-6
-2
x
three geophysical models and a specquation. As obserand squared wind
x
(a) τ = 0.25
(a) τ = 0.25
10
10
8 6
8
4
6
2
4
(6)
z
z
odel for Rayleigh– 63):
0
0, ρ = 28, and β =
0
Fig. 5: Con zonal regim barotropic v tions of the t
2
-2
0
-4
= x2 + y 2 + z 2
x
-2 x
(b) τ = 0.5
of convection and x in (5) we respec-
(b) τ = 0.5
4
100 0 1 0 001
6 2 6
4
z
ample of N = 10 requency ω = 100. vables evaluated at
0
e set Λτ,q changes servable φcon . For -2 al conditions leadx le have a negative (c) τ = 1.5 of convection are For τ > 0.25 more 3: Projection of the Lorenz–63 attractor on the (x,z)leading to an Fig. ex- 2. Fig. Projection of the Lorenz-63 attractor on the (x, z)-plane plane with colours indicating the magnitude of the maximal LE which is larger with colours indicating the magnitude of the maximal time-τ FTLE. time-τ FTLE. Black dots mark initial conditions leading to s that extreme valBlack dots mark initial conditions leading to observable an exceedance an exceedance of the 95th percentile of the φcon . of When the lead time the 95thFor percentile of theconditions observable φconto. For τ = typically 0.25, initial τ = 0.25 initial leading extremes ifferent thresholds conditions to extremes typically low have FTLE, but for haveleading low FTLE, but for τ = 0.25 theyhave typically a large difference in preFTLE. This difference vanishes for longer lead times. τ = 0.25, they typically have a large FTLE. This difference vanemes dissappears.
ishes for longer lead times. e set Λτ,q changes for the observable
By numerical integration we computed a sample of N = 106 points on the attractor using the sample frequency ω = 100. Table 1 lists some statistics of the observables evaluated at the sample points. Figure 1 shows how the distribution of the set 3τ,q changes with lead time and threshold q for the observable φcon . For τ ≤ 0.25 more than 75 percent of the inital conditions leading to an exceedance of the 99th percentile have a Nonlin. Processes Geophys., 19, 529–539, 2012
z
z˙ = xy − βz,
A.E. Sterk et al.: Predictability of extreme values in geophysical models
2
are systemat Only for lea tions of Λτ, shown). Fig. 3 and φen the FTL various lead which lead t stated lead t large FTLE where the tw each other. regions quic conditions l coincide wit whereas init not.
0 x (c) τ = 1.5
Fig. 4: As Fig. 3, but for the observable φ . Initial condi-
en Fig. 3. As Fig. 2, but for the observable φen . Initial conditions leadtions leading to extremes typically also have a large FTLE, ing to extremes typically also have a large FTLE, and this persists and this persists for larger lead times. for larger lead times.
φen . For all lead times τ almost all initial conditions leading to an exceedance of the 95th percentile of the energy have a FTLE which is larger than the median of Λτ,−∞ . This means negative FTLE. suggestvalues that of extreme values of convecthat in this This case extreme the energy observable φen
tion are well-predictable up to small lead times. For τ > 0.25 more than 75 percent of the initial conditions leading to an exceedance of the 99th quantile have a FTLE that is larger than the median of 3τ,−∞ , which indicates that extreme values for this lead time are less predictable. When the lead time τ increases, the distributions of 3τ,q for different thresholds q become similar; this suggests that the difference in predictability between extremes and non-extremes disappears. www.nonlin-processes-geophys.net/19/529/2012/
3.2 The ba
In this sectio (BVE) for at
∂ ∆ψ ∂t
where ψ is t Coriolis forc erator. In ad
A. E. Sterk et al.: Predictability of extreme values in geophysical models A.E. Sterk et al.: Predictability of extreme values in geophysical models
0%
95%
97%
3
533
99%
12
Λτ,q
8 4 0 -4 0.25
0.5
0.75
1
1.25
1.5
1.75
τ
Fig. 2: As Fig. 1, but for the observable φen .
Fig. 4. As Fig. 1, but for the observable φen .
A.E. Sterk et al.: Predictability of extreme values in geophysical models
z
z
the direction and magnitude of maximal error growth along a finite-time segment of an orbit in state space (Buizza and Figure 4 shows howPalmer, the distribution setLeutbecher 3τ,q changes 1995; Palmer etof al.,the 1998; and Palmer, 12 observable with increasing lead2008). time and threshold for the To measure the growth of perturbations over a time interφen . For all lead times τ almost all initial conditions leading val [0,τ ] we use different norms at t =10 0 and t = τ . Let C be to an exceedance ofthe theinitial 95th percentile of the energy a error covariance matrix, and let P be have an orthogoprojection matrix (i.e., = P2 =8 P. ⊤This ). Let means k · k denote FTLE that is larger nal than the median ofP3 τ,−∞ the Euclidean norm. The Rayleigh–Ritz quotient
that in this case extreme values of the energy observable φen 2 kP Lτ (x)vk v ⊤ Lτ (x)⊤6P Lτ (x)vvalues. are systematically less predictable than (5) = non-extreme v ⊤ C −1 v kC −1/2 vk2 Only for lead times much longer than τ = 1.75, the distribu4 is maximalquantiles if and only ifq vbecome is the rightsimilar singular vector tions of 3τ,q for different (not of P Lτ (x)C 1/2 associated with the largest singular value. We shown). define the time-τ finite-time Lyapunov2exponents (FTLE) as Figures 2 and 3 show how for the observables φcon and 1 λi (x,τ ) = parts log σi (x,τ 1,...,n, φen the FTLEs vary on different on ),thei0=attractor for τ various lead times. where Black dots mark the initial conditions σ1 (x,τ ) ≥ σ2 (x,τ ) ≥ ··· ≥ σn-2 (x,τ ) are the singular which lead to an exceedance the1/295th percentile thethe values of P Lof . In the following wewithin assume that τ (x)C x error covariance C isinitial the identity matrix. stated lead time. Forinitial short lead times, conditions with choose the projection matrix P such that the singular (a)Weτ in = 0.25 large FTLE are located the central regionofof the attractor vectors will point in the direction maximal growth of the observable For an observable of theattractor form (2) wecross set where the two wings of theφ.butterfly-shaped ⊤ each other. When the lead time increasesqqthe unpredictable 10 P= kqk2 regions quickly spread out on the entire attractor. The initial conditions leading to extreme energy values systematically 8 so that φ(x)2 larger FTLEs, 2 coincide with the parts on the attractor . kP xkhaving = 2 kqk whereas initial conditions leading to extreme convection do 6 In case of observable (3) recall that a symmetric nonnegnot. ative definite n × n matrix has k ≤ n positive eigenvalues 4 3.2 The barotropic vorticity equation 2 In this section we consider the barotropic vorticity equation (BVE) for atmospheric flow over orography: 0 ∂ 1ψ = −J (ψ, 1ψ + βy + γ h) − C1(ψ − -2 ψ ∗ ), ∂t x
(7)
∗ where ψ is the stream (b) function, τ = 0.5 ψ is the forcing, βy is the Coriolis force, h is the orography, and 1 is the Laplace operator. In addition, J is the Jacobian operator: 6
z
4
www.nonlin-processes-geophys.net/19/529/2012/ 2
5
µ1 ≥ ··· ≥ µk > 0 and the corresponding eigenvectors qi are orthonormal. We can rewrite (3) as
πb
φ(x) = x⊤ Qx =
k X
µi (qi⊤ x)2 .
i=1
In this case we set P=
k X
qi qi⊤
i=1
so that kP xk2 = and
0
k X
(qi⊤ x)2 ,
i=1
µk kP xk2 ≤ φ(x) ≤ µ1 kP xk2 .
2π
Hence, the φ(x) is large if and only if kP xk is large.
πb
2.3 Predictability of extreme values We fix an initial point x0 on the attractor and a sampling frequency ω > 0 and compute a sample of points along an orbit on the attractor: S = { Φk/ω (x0 ) | k = 1,...,N }. We choose ω and N sufficiently large to provide a good sampling of the attractor as far as both local and global fluctuations are concerned. Typically, we take N = 106 , whereas ω depends 0 on the time scale of the system of interest. To study the predictability of extreme events we define the set ) | x ∈ Sof and the Φτ (x)stream ∈ Eq } τ,q = {λ1 (x,τplots Fig. 5: ΛContour function
2π
fields of the Fig. 5. Contour plots of the stream function fields of the zonal containing(top) the FTLEs for initial conditions such that after regime and blocked regime (bottom) in the(bottom) barotropic in vor-the zonal regime (top) and blocked regime τ units of time the observable exceeds the threshold q. We ticity Eq. (8). Blue dots mark the (8). locations of dots the three wind speed barotropic vorticity equation Blue mark the locaobservables (see main text). tions of the three wind speed observables (see main text).
are systematically less predictable than non-extreme values. Only for lead times much longer than τ = 1.75 the distribu∂f ∂g ∂f ∂g tions J (f,of g) Λ =τ,q for different − . quantiles q become similar (not ∂x ∂y ∂y ∂x shown). Fig. 3 and Fig. 4 show how for the observables φcon and The domain is [0, 2π]×[0, π b], where b equals twice the asφen the FTLEs vary on different parts on the attractor for pect ratio of the β–plane channel. We take periodic boundary various lead times. Black dots mark the initial conditions conditions in x and for y = 0, π b, we require which lead to an exceedance of the 95th percentile within the stated lead time. For short lead times initial conditions with Z2π large located in the central region of the attractor ∂ψ FTLE are ∂ψ = 0, dx = 0, where of the butterfly-shaped attractor cross ∂x the two wings ∂y 0 each other. When the lead time increases the unpredictable regions quickly spread out on the entire attractor. The inital Nonlin.to Processes 19, 529–539, 2012 conditions leading extremeGeophys., energy values systematically coincide with the parts on the attractor having larger FTLEs, whereas initial conditions leading to extreme convection do
534
A. E. Sterk et al.: Predictability of extreme values in geophysical models
Table 2. Parameters and coefficients of the model Eq. (8) as used by Crommelin et al. (2004). 6
Par
Value
Par
Par C b ψˆ1∗
C b ψˆ ∗
r ψˆ4∗ α1
r ψˆ ∗
0.1 Value 0.1 0.5 0.5 0.95 0.95 -0.801 -0.801 ˆ∗∗ rˆψ rψ √1 √ 8 12b22/3π(b2 +21) 8 2b /3π(b + 1)
α2Par β α2 β β1β1 β2β2 δ1δ1 δ2 δ2
1 4
α1
ValueA.E. Sterk et al.: PredictabilityPar Value of extreme values in geophysical models √ 2 32 2(bValue + 3)/15π(b2 + 4) γ 0.2 Par Value √ √ 2 32 2(b2 + 3)/15π(b2 + 4) 0.2 2bγ /3π(b 5/4 γ1 γ4 √ √ + 1) 2 5/4 γ1 4 √2bγ/3π(b 2 2 2 + 4)+ 1) βb /(bβb+ γ2 γ32 2bγ /15π(b 2 1) 2bγ/15π(b2 + 4) /(b2 + 1) 32 2√ √ 2 2 2 βb√/(bβb√ + 4)2 + 4) γ˜1 γ˜41 √2bγ /3π4√2bγ/3π /(b 2 2 /15π(b 2 +21) + 1) 8 √2bγ/15π 64√2b64 γ ˜2 γ˜82 √ 2bγ /15π √2b 2/15π(b 64 2(b − 3)/15π(b2 + 4) ε 16 2/5π 64 2(b2 − 3)/15π(b2 + 4) ε 16 2/5π
Table 2: Parameters and coefficients of the model (8) as used by Crommelin et al. (2004).
0%
95%
97%
99%
0.5 0.4
Λτ,q
0.3 0.2 0.1 0 -0.1 0.5
1
1.5
2
2.5
τ
Fig. 6. As Fig. 1, but for the model Eq. (8) with observable φsid .
Fig. 6: As Fig. 1, but for the model (8) with observable φsid .
The domain [0,2π] [0,πb], where boundaries b equals twice the which respectively mean isthat on×the lateral theaspect ratio of the β–plane channel. We take periodic boundary northward component of the velocity vanishes and that the conditions in x and for y = 0,πb we require net zonal flow is zero. Z ∂ψstudied a2π3-dimensional ∂ψ Charney and DeVore (1979) spec= 0, dx = 0, ∂x ∂y 0 tral truncation of Eq. (7), and detected coexisting equilibrespectively mean flow that onpatterns. the lateral De boundaries ria representing which blocked and zonal Swartthe component of the velocity vanishes and (1989) studied northward higher order truncations, including thethat6-the net zonal flow is zero. dimensional truncation Charney and DeVore (1979) studied a 3-dimensional spectral truncation of (7), and detected coexisting equilibria rep˙ ψˆ = γ˜ ψˆ − C(ψˆ − ψˆ ∗ ), 1
1 3
1 resenting blocked and zonal flow patterns. De Swart (1989) 1 studied higher order truncations, including the 6-dimensional ˙ ˆ ˆ ˆ ˆ ψ2 = −(α1 ψ1 −truncation β1 )ψ3 − C ψ2 − δ1 ψˆ 4 ψˆ 6 ,
˙ ˆ 1C( ˆ 4 ψˆ 5 , −ψˆC ψˆψˆ31∗+ ψ˙ˆ 3 = (α1 ψˆ 1 − β1 )ψ − ), δ1 ψ ψˆ12=−γ˜1γψˆ13ψ 1− ˙ ˆ ˆ ˆ ˆ ψˆ2 ˆ=∗−(α1 ψˆ1 − 1ˆ)ψ3 − C ψ˙ˆ = γ˜ ψˆ − C(ψˆ − ψ ) + ε(ψˆ βψ − ψˆψ2ψˆ− δ),1 ψ4 ψ6 , 4
2 6
(8)
4
2 6 3 5 4 ˙ ψˆ3 = (α1 ψˆ1 − β1 )ψˆ2 − γ1 ψˆ1 − C ψˆ3 + δ1 ψˆ4 ψˆ5 , ˙ (8) ψˆ 5 = −(α2 ψˆ 1 − β2 )ψˆˆ˙ 6 − Cˆψˆ 5 − ˆδ2 ψˆ ˆ4∗ψˆ 3 , ˆ ˆ ˆ ˆ ψ4 = γ˜2 ψ6 − C(ψ4 − ψ4 ) + ε(ψ2 ψ6 − ψ3 ψ5 ), ˆ˙ ˆ ψˆˆ2ψ.ˆ , ψ˙ˆ 6 = (α2 ψˆ 1 − β2 )ψ γ2 2ψˆψˆ41 − −βC ψ ˆˆ + δˆ5 ψ − δ42ψ ψˆ55=−−(α 4 3 2 )ψ66− C ψ2 ˙ ˆ1 − β2 )ψˆ5 − γ2of ψˆ4Crommelin − C ψˆ6 + δ2 ψˆ4 ψˆet ψˆ6 = the (α2 ψ We study this model with parameters 2 . al.
(2004), which are listed in Table 2. With this choice the channel [0, 2π] × [0, πb] has dimensions 5000 × 1250 km, the orography h(x, y) = cos(x) sin(y/b) has a dimensional amplitude of 200 m, and one unit of time corresponds to one day. Crommelin et al. (2004) found intermittent transitions between two regions in state space corresponding to zonal and blocked flows, see Fig. 5. The dynamics consists of three recurrent episodes: (1) transitions from zonal to blocked flows, Nonlin. Processes Geophys., 19, 529–539, 2012
We study this model30 with the parameters of Crommelin al. typically taking days, (2) transitions frometblocked to zonal (2004), which are listed in Table 2. With this choice the chanflows, typically taking 40–80 days, and (3) spiraling benel [0,2π]×[0,πb] has dimensions 5000×1250 km, the oroghaviour around the zonal regime, typically lasting more than raphy h(x,y) = cos(x)sin(y/b) has a dimensional amplitude of 200days. m, andThis one unit of time corresponds to one day. 200 intermittent behaviour can be explained by al. (2004) found intermittent transitions betheCrommelin presenceetof Shil’nikov-like strange attractors, due to hotween two regions in state space corresponding to zonal and moclinic bifurcations taking place near a Hopf-saddle-node blocked flows, see Fig. 5. The dynamics consists of three recurrent episodes: (1) transitions from zonal to blocked bifurcation (Broer and Vegter, 1984; Broer flows, and Vitolo, 2008). typically taking 30 days, (2)use transitions from blocked to zonal As observable we the squared wind speed at several flows, typically taking 40–80 days, and (3) spiraling belocations in the β-plane channel. In Appendix haviour around the zonal regime, typically lasting more than A we show 200 days. This and intermittent behaviour can be explained that the eastnorthward components of thebywind vector at the y) presence of Shil’nikov-like strangethe attractors to ho(x, can be computed from state due vector ψˆ as q1> ψˆ and moclinic bifurcations taking place near a Hopf-saddle-node > ˆ 6 are orthogonal vectors depending qbifurcation (Broerqand Broer and Vitolo, 2008). 1 , qVegter, 2 ∈ R1984; 2 ψ, where As observable we use the squared wind speed at several on (x, y). Hence, the squared wind speed at location (x, y) is locations in the β-plane channel. In Appendix A we show given by that the east- and northward components of the wind vector ˆ
⊤
ˆ
> ˆthe2state ˆvector > ψ as >q1 ψ and> ˆ at (x,y) ˆ =can(qbe1>computed ˆ 2 + (qfrom φ( ψ) ψ) 2 ψ) = ψ (q1 q1 + q2 q2 )ψ. ⊤ˆ 6
q2 ψ, where q1 ,q2 ∈ R are orthogonal vectors depending on (x,y). the squared windinspeed location (x,y) is we take For theHence, projection matrix Eq.at(5) given by q1ˆq > ⊤ ˆq22 q > ⊤ ˆ 2 ˆ⊤ ⊤ ⊤ ˆ P = φ(ψ) 1=2(q+1 ψ) +2(q22. ψ) = ψ (q1 q1 + q2 q2 )ψ. kq1 k kq2 k
We will study the extreme wind speeds at three locations: the side of the orography (x, y) = (0, 3π b/4), the top (x, y) = (0, π b/2), and the valley (x, y) = (π, π b/2), see Fig. 5. The corresponding wind speed observables are denoted by φside , φtop , and φval . By numerical integration we computed a sample of N = 106 points on the attractor using the sample frequency ω = 2. Table 1 lists some statistics of the observables evaluated at the sample points. www.nonlin-processes-geophys.net/19/529/2012/
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τ1.5
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2 2
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Fig. 7. As Fig. 6, but for the observable φtop .
τ
Fig. 7: As Fig. 6, but for the observable φtop . Fig. 7: As Fig. 6, but for the observable φtop .
0.5
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1 1
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Fig. 8. As Fig. 6, but for the observable φval .
Fig. 8: As Fig. 6, but for the observable φval . Fig. 8: As Fig. 6, but for the observable φval .
For the projection matrix in (5) we take For the projection matrix in ⊤ (5) we take q2 q2⊤ q1 q1 + . P =the distributions Figures 6, 7, and 8 show how of the set kq q22qk2⊤2 q11qk1⊤2 kq + . for three obP = threshold 3τ,q change with lead time τ and q 2 2 kq k kq2 k will study the extreme 1wind speeds three locations: servables φsid , φWe φ orography . For φ(x,y) initialat conditions top , and sid the the the val = (0,3πb/4), the toplocations: (x,y) = Weside willofstudy the extreme wind speeds at three leading to extremes systematically a= large FTLE, which (0,πb/2), valleyhave (x,y) (π,πb/2), see 5. The the side ofand the the orography (x,y) = (0,3πb/4), the Fig. top (x,y) = corresponding speed(x,y) observables by5.φside means that extremes are and lesswind predictable. φare the FTLEs (0,πb/2), the valley =For (π,πb/2), see Fig. The, topdenoted φcorresponding φval .wind Byextremes numerical integration wevariable. computed top , and speed observables are denoted by φsidea, for initial conditions leading to are highly 6 sample of Nφval = .10By points on theintegration attractor using the samplea φtop , and numerical we computed In this case onefrequency cannot say that extremes are either better 6 ==2.10 Table 1 lists some statistics of the observsample of ωN points on the attractor using the sample ables evaluated at the sample or less predictable than generic events. Finally, forofφthe the frequency ω = 2. Table 1 listspoints. some statistics val observFig.evaluated 6, 7, and at 8leading show how distributions the set Λτ,q the sample points. FTLEs for initialables conditions tothe extremes areofsystemchange τ and q for three Fig. with 6, 7, lead and 8time show howthreshold the distributions of observables the set Λ atically low: more than 75% of the initial conditions leadingτ,q change with lead time τ and threshold q for three observables
to extremes have an FTLE smaller than the median FTLE for all events. Figures 9–11 show the FTLEs vary on different parts of the attractor. For all observables the initial conditions with large FTLE are located near the zonal regime and the initial conditions with small FTLE are located near the blocked regime. This separation persists remarkably for increasing lead time. This is very different from the Lorenz-63 attractor, for which the unpredictable regions quickly spread out along the attractor. For the observables φsid and φtop the initial conditions leading to extreme wind speeds systematically coincide with
www.nonlin-processes-geophys.net/19/529/2012/
φsid , φtop , and φval . For φsid the initial conditions leading to extremes systematically a large FTLE, which means φsid , φtop , and φval . For φhave sid the initial conditions leading that extremes are less predictable. For FTLE, φattractor. FTLEs forφ top the to extremes systematically haveof a large which means the unpredictable parts the For val the initial initial conditions to extremes areφhighly variable. In that extremes areleading less predictable. For top the FTLEs for conditions leading to extremes intersect the more predictable this case one cannot say that extremes are are highly either better or less initial conditions leading to extremes variable. In regions onthan thegeneric attractor. predictable events. Finally, φvalbetter the FTLEs this case one cannot say that extremes are for either or less for initial conditions leading to extremes areφsystematically predictable than generic events. Finally, for val the FTLEs low: moreconditions than 75% leading of the initial conditions leading to exfor initial to extremes are systematically tremes havethan an FTLE smaller than conditions the medianleading FTLE for all 3.3 The Lorenz-96 low: more 75% of the model initial to exevents. tremes have an FTLE smaller than the median FTLE for all events. Figures 9–11 show the FTLEs vary on different parts of the
The Lorenz-96 modelthe is initial a conceptual model for traveling attractor. allshow observables withoflarge FiguresFor 9–11 the FTLEs vary onconditions different parts the waves in the atmosphere (Lorenz, 1996). FTLE are For located near the zonal and the initial condiattractor. all observables the regime initial conditions withThis large model is used tions with small assimilation FTLE arezonal located nearand the the blocked FTLE aredata located near the regime initialregime. condi- and Palatella, to test methods (Trevisan tions with small FTLE are located near the blocked regime. 2011) and subgrid scale parameterizations (Crommelin and Vanden-Eijnden, 2008). Although the model is not derived from physical principles it has features which are typical for geophysical models: forcing, dissipation, and energy preserving quadratic terms. The model variables x1 , . . . , xn can be interpreted as meteorological quantities, such as pressure or vorticity, along a circle of constant latitude where the index i of each variable xi plays the role of longitude. The dynamical equations are dxi = xi−1 (xi+1 − xi−2 ) − xi + F, dt
i = 1, . . . , n > 3, (9)
Nonlin. Processes Geophys., 19, 529–539, 2012
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A. E.A.E. Sterk et al.: of extreme values in geophysical models etetPredictability al.: ofofextreme values iningeophysical models A.E.Sterk Sterk al.:Predictability Predictability extreme values geophysical models
0.3 0.3
0.5 0.5 0.4 0.4
0.2 0.2 ψˆ1 ψˆ1
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ψˆ1 ψˆ1
(a)(a) τ= 0.50.5 τ=
0.2 0.2
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00
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00
ψˆ3ψˆ3
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(b)(b) τ= 2.52.5 τ=
(b)(b)τ τ== 2.5 2.5
bψb,1ψb)-plane bthe Fig. Projection (bψb Fig.9: Projection ofa athe the attractor Fig. 9. 9: Projection on on theon (the ψ of a theof attractor ofattractor Eq. (8) 3ψ, 3 1 )-plane 3, ψ 1()-plane colour indicating the magnitude ofmagnitude the maximal time-τ FTLE. ofwith with colour thethemagnitude ofof the of(8)(8) with colourindicating indicating themaximal maximal The grey circleThe (square) indicates the pointindicates inindicates phase space corretime-τ FTLE. grey circle the time-τ FTLE. The grey circle(square) (square) thepoint pointinin sponding withcorresponding the zonal (blocked) regime ofzonal Fig. (blocked) 5.(blocked) Black dotsregime mark phase with thethezonal phasespace space corresponding with regime initial conditions leading to an exceedance of the 95th percentile of ofof Fig. 5. 5. Black dots mark initial conditions leading totoananexFig. Black dots mark initial conditions leading exthe observable φsid . Note that initial conditions leading to high wind ceedance of the 95th percentile of the observable φ . Note ceedance of the 95th percentile of the observable φ . Note sidsid are speeds are located near the zonal regime, where also the FTLEs that initial conditions leading high wind speeds are located that initial conditions leading toto high wind speeds are located large. near zonal regime, where alsothethe FTLEsare arelarge. large. near thethe zonal regime, where also FTLEs
Fig. 10: 9,9,but . In case initial Fig. 10: AsFig. Fig. butfor forobservable observable φthis .case Inthis this case initial top Fig. 10. AsAs Fig. 9, but for observable φtop . φ Intop initial conconditions wind are confined conditions leading tolarge large wind speeds arenot not confined ditions leadingleading to large to wind speeds arespeeds not confined to the unpre- toto the dictable region of the region attractor. theunpredictable unpredictable regionofofthe theattractor. attractor.
with periodic “boundary conditions” xi+n = xi . The dimenThis persists remarkably increasing lead time. This separation persists forforincreasing lead sion nseparation and forcing F > remarkably 0 are free parameters. We set n =time. 36 This is very different from the Lorenz–63 attractor, for which This is very different from the Lorenz–63 attractor, for which and F = 8, which were originally used by Lorenz. unpredictable regions quickly spread out along the attracthethe unpredictable regions quickly spread out along the attracWe use the observable tor.For Forthetheobservables observablesφsid φsidand andφtop φtopthetheinitial initialconditions conditions tor. nextreme leading wind speeds systematicallycoincide coincidewith with leading toto extreme wind speeds systematically X φthe (x) = xi2 parts unpredictable partsofofthetheattractor. attractor.For Forφval φvalthe theinitial initial enunpredictable the i=1 conditions leading extremes intersect more predictable conditions leading toto extremes intersect thethe more predictable regions attractor. regions onon thethe attractor. which measures the total energy of the system. For the projection inLorenz–96 Eq. (5), wemodel take the identity matrix. By numeri3.3 The The Lorenz–96 model 3.3 cal integration we computed a sample of N = 106 points on the attractor using the is sample frequencymodel ω = 100. Table 1 TheLorenz–96 Lorenz–96 model isa aconceptual conceptual model traveling The model forfortraveling waves atmosphere (Lorenz, 1996).This Thismodel modelis isused used waves inin thethe atmosphere (Lorenz, 1996). totest testdata data assimilation methods (Trevisan and Palatella, toNonlin. assimilation methods (Trevisan and Palatella, Processes Geophys., 19, 529–539, 2012
2011) and subgrid scale parameterizations and 2011) and subgrid parameterizations lists some statistics ofscale the observable evaluated(Crommelin at(Crommelin the sampleand Vanden-Eijnden,2008). 2008).Although Althoughthe themodel modelisisnot notderived derived Vanden-Eijnden, points. fromphysical physical principles has featureswhich which are typical for from principles it ithas features are typical Figure 12 shows how the distribution of the set 3τ,q for geophysical models: forcing, dissipation, and energy pregeophysical models: dissipation, energy changes with lead time τforcing, and threshold q. For and all lead timespreτserving , the distribution ofterms. the maximal FTLE does not change very serving quadratic terms. quadratic much with increasing threshold q. This suggests that extreme The model variables can interpreted meThe model variables x1x,...,x bebeinterpreted asasme1 ,...,x n ncan values of the quantities, energy observable neither better nor along less teorological quantities, suchasasare pressure vorticity, alonga a teorological such pressure oror vorticity, predictable than non-extreme values. circleofofconstant constant latitude,where wherethe theindex indexi of i ofeach eachvariable variable circle latitude, plays the role of longitude. The dynamical equations are xixplays the role of longitude. The dynamical equations are i
dxdx i i Conclusions and discussion xi−1 − xi−2 )− i =1,...,n 1,...,n>>3,3, (9) (9) == xi−1 (x(x − xi−2 )− xix+ F,F, i = i+1 i+ i+1 dtdt Inwith thisperiodic paper we investigated the predictability extreme with periodic ‘boundary conditions’ xi+n==xixof Thedimendimen‘boundary conditions’ xi+n .i .The values in geophysical models. We studied how FTLEs sionn nand andforcing forcingFF>>0 0are arefree freeparameters. parameters.We Weset setnden==3636 sion pend lead time and the threshold on the obandFonF=forecast = which were orginally used Lorenz. and 8 8which were orginally used bybyLorenz. servable. General statements on whether extreme values are Weuse use theobservable observable We the better or worse to predict are not possible. Whether initial n nX X φen (x)== x2ix2i φen (x) i=1 www.nonlin-processes-geophys.net/19/529/2012/ i=1 4
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TableModel 3. The predictability of extreme values for the three models Obs. Predictability of extremes with different observables. Lorenz–63 φ For lead times up to τ = 0.25 extremes con
Model
ψˆ1
0.4
Lorenz-63
0.3 BVE
0.2
are well-predictable (negative FTLEs), Predictability of extremes but for longer lead times they are just as as to non-extremes. φcon For predictable lead times up τ = 0.25 extremes are well-predictable (negative FTLEs), φen For lead times up to τ = 01.75 extremes but for longer lead times they are just as are less predictable than non-extremes. predictable as non-extremes. φsid Extremes are less predictable, and this φen For persists lead times τ = lead 01.75times extremes upup totolong due to the are less predictable thanofnon-extremes. intermittent nature the dynamics. Obs.
φsidφtopExtremes are are lessbetter predictable, and this Extremes predictable than nonpersists up to long lead times due to the extremes. intermittent nature of the dynamics. φval Extremes are neither better nor less preφtop Extremes arethan neither better nor less dictable non-extremes. predictable than non-extremes. Lorenz–96 φen Extremes are neither better nor less predictable non-extremes. φ Extremes arethan better predictable than
BVE
ψˆ3
0.1
(a) τ = 0.5
0.4
val
non-extremes
Table 3: The predictability of extreme values for the three Lorenz-96 Extremes are neither better nor less models withφen different observables. predictable than non-extremes.
ψˆ1
0.3
0.2
ψˆ3
0.1
(b) τ = 2.5 Fig. Fig. 9, but for for observable φval . φ Inval this case Fig.11.11:AsAs Fig. 9, but observable . In thisinitial caseconinitial ditions leadingleading to large to wind speeds arespeeds located are in the more preconditions large wind located in the dictable regions of theregions attractor.of the attractor. more predictable
which measures the total energy of the system. For the projectionleading in (5) to weextreme take the identity By numericonditions values havematrix. larger FTLEs depends on (1) the observable, (2) the attractor of the cal integration we computed a sample of N = 106system, points on and the prediction lead time. frequency Table 3 presents the main the (3) attractor using the sample ω = 100. Table 1 conclusion for each model/observable pair. lists some statistics of the observable evaluated at the sample FTLEs measure forecast error growth rate under the aspoints. sumption errors initial condition are ΛinfinitesiFig. 12that show howinthethedistribution of the set τ,q changes mally small. For finite-size, but small, errors, the maximal with lead time τ and threshold q. For all lead times τ the disFTLE still of is the a good estimate of error (Harle et al., tribution maximal FTLE does growth not change very much 2006). For larger errors, however, error growth may no longer with increasing threshold q. This suggests that extreme valbeues exponential and this will alsoareeffect the better predictability of the energy observable neither nor lessofpreextreme values. The present study should, therefore, be exdictable than non-extreme values. tended to the setting of ensemble forecasts in which the predictability of extremes is measured in terms of the dispersion of4 ensemble members. approach would also be more apConclusions andThis discussion propriate for operational weather forecasting models without a In tangent linearwe model needed tothe compute FTLEs, of such as this paper investigated predictability extreme values in geophysical models. We studied how FTLEs depend on forecast lead time and the threshold on the observwww.nonlin-processes-geophys.net/19/529/2012/
able. General statements on whether extreme values are betthe MOGREPS system of the UK Met Office (Bowler et al., ter or worse predictable are not possible. Whether initial con2008). ditions leading to extreme values have larger FTLEs depends Another important question is: how predictable are realon (1) the observable, (2) the attractor of the system, and (3) world extremes, such as wind storms? Large-scale flow patthe prediction lead time. Table 3 presents the main concluterns, such as the North Atlantic Oscillation, cause temposion for each model/observable pair. ral clustering of storms (Mailier et al., 2006; Vitolo et al., measure of forecast error growth the as2009).FTLEs The emergence these patterns might rate be a under manifessumption that errors in the initial condition are infinitesitation of intermittency, i.e. the irregular alternation between mally butsuch small, errors the maximal phases of small. chaoticFor andfinite-size, non-chaotic, as steady or periFTLE still is a good estimate of error growth (Harle odic dynamics (Pomeau and Manneville, 1980). For exam-et al., For larger errors, growth may no ple,2006). the spectral truncation Eq. however, (8) of the error barotropic vorticlonger be exponential and this will also effect the predictability equation exhibits intermittent transitions between zonal of extreme present study shouldattractors therefore be andity blocked flowsvalues. due to The Shil’nikov-like strange extended to the setting of ensemble forecasts in which the appearing near a Hopf-saddle-node bifurcation (Crommelin predictability is measured terms 1984). of the diset al., 2004; Broerof andextremes Vitolo, 2008; Broer andinVegter, persion members. approach also be Other formsofofensemble intermittency due to This bifurcations of would planetary more appropriate for operational weather forecasting models waves have been detected in low-order models of the shala tangent(Sterk linear etmodel needed to compute FTLEs, lowwithout water equations al., 2010). Because different intermittent phases can have different error growth rates, the such as the MOGREPS system of the UK Met Office (Bowler emergence of large-scale flow patterns might enhance preet al., 2008). dictability of Anotherextremes. important question is: how predictable are realFinally, we note such that asFTLEs might not be the flow best patworld extremes, wind storms? Large-scale measures of finite-time predictability. A different approach terns, such as the North Atlantic Oscillation, cause tempowould be to applyoftechniques based on reconstrucral clustering storms (Mailier et Takens’ al., 2006; Vitolo et al., tion2009). theorem, such as correlation integrals and entropy; The emergence of these patterns might be see a maniBroer and Takens (2011) and references therein. Such tech- befestation of intermittency, i.e., the irregular alternation niques have been of applied to and develop early warning tween phases chaotic non-chaotic, such systems as steady or for periodic, thermal excursions in chemical reactors (Zald´ var et al., dynamics (Pomeau and Manneville, ı1980). For ex2005). Potentially, these techniques can be useful in the preample, the spectral truncation (8) of the barotropic vorticdiction of extreme values in geophysicaltransitions applications. We be-zonal ity equation exhibits intermittent between lieve that these questions and problems will have sufficient and blocked flows due to Shil’nikov-like strange attractors potential for future research. appearing near a Hopf-saddle-node bifurcation (Crommelin et al., 2004; Broer and Vitolo, 2008; Broer and Vegter, 1984). Other forms of intermittency due to bifurcations of 2012 planetary Nonlin. Processes Geophys., 19, 529–539,
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0%
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8 6 4 2 0 0.2
0.4
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0.8
1
τ
Fig. 12: As Fig. 1, but for the Lorenz–96 model (9) with observable φen .
Fig. 12. As Fig. 1, but for the Lorenz-96 model Eq. (9) with observable φen . waves have been detected in low-order models of the shallow water equations (Sterk et al., 2010). Because different intermittent phases can have different error growth rates, the emergence of large-scale flow patterns might enhance predictability of extremes.
Appendix A
Wind speed in the barotropic vorticity equation
Finally, we note that FTLEs might not be the best measures of finite-time predictability. A different approach apply techniques based Takens’ reconstrucThe stream functionwould can be betocomputed from theonspectral coeftion theorem, such as correlation integrals and entropy; see ficients ψˆ i by Broer and Takens (2011) and references therein. Such techniques have been applied to develop early warning systems √ √ in chemical reactors (Zald´ıvar et al., for thermal excursions ψ(x, y) = b 2ψˆ 1 cos(y/b) + b these 2ψˆ 4techniques cos(2y/b) 2005). Potentially, can be useful in the prediction of extreme values in geophysical applications. We + 2b{ ψˆ 2 cos(x) ψˆ 3 sin(x)} sin(y/b) believe that these+ questions and problems will have sufficient potential for future research.
+ 2b{ψˆ 5 cos(x) + ψˆ 6 sin(x)} sin(2y/b).
De Swart (1988) uses the convention u = in−ψ v = ψx . vorticity This Appendix A Wind speed they ,barotropic equation gives the following expressions for the east- and northward components of the velocity field: The stream function can be computed from the spectral coefˆ
ficients ψi by √ √ u(x, y) = 2ψˆ 1 sin(y/b) + 2√ 2ψˆ 4 sin(2y/b) √
ψ(x,y) = b 2ψˆ1 cos(y/b) + b 2ψˆ4 cos(2y/b)
ˆ sin(x)} − 2{ψˆ 2 cos(x) + ψ cos(y/b) +3 2b{ψˆ cos(x) + ψˆ sin(x)}sin(y/b) 2
3
ˆ 62b{sin(x)} + ψˆ5 cos(x)cos(2y/b), + ψˆ6 sin(x)}sin(2y/b). − 4{ψˆ 5 cos(x) + ψ De Swart (1988) uses the convention u = −ψy ,v = ψx . This
gives the− following expressions for the east- and northward v(x, y) = −2b{ψˆ 2 sin(x) ψˆ 3 cos(x)} sin(y/b)
− 2b{ψˆ 5 sin(x) − ψˆ 6 cos(x)} sin(2y/b). This can be rewritten as u(x, y) = q1> ψˆ
ˆ and v(x, y) = q2> ψ,
where
√
2 sin(y/b) 2 cos(x) cos(y/b) 2 sin(x) cos(y/b) √ q1 = 2 2 sin(2y/b) −4 cos(x) cos(2y/b) −4 sin(x) cos(2y/b) Nonlin. Processes Geophys., 19, 529–539, 2012
components of the velocity field: andu(x,y) = √2ψˆ1 sin(y/b) + 2√2ψˆ4 sin(2y/b)
− 2{ψˆ2 cos(x) + ψˆ3 sin(x)}cos(y/b) 0 − 4{ψˆ5 cos(x) + ψˆ6 sin(x)}cos(2y/b),
−2b sin(x) sin(y/b) 2b cos(x) sin(y/b) . = −2b{ψˆ2 sin(x) − ψˆ3 cos(x)}sin(y/b) q2 v(x,y) = 0 − 2b{ψˆ5 sin(x) − ψˆ6 cos(x)}sin(2y/b). −2b sin(x) sin(2y/b) This can be rewritten as 2b cos(x) sin(2y/b) ˆ u(x,y) = q1⊤ ψˆ and v(x,y) = q2⊤ ψ,
Note q and q2 are orthogonal. where that the vectors √ 1
2sin(y/b) 2cos(x)cos(y/b) 2sin(x)cos(y/b) √ q1 = 2This Acknowledgements. research 2sin(2y/b) is part of the project “PRE −4cos(x)cos(2y/b) DEX: Predictability of Extreme Weather Events”, funded by −4sin(x)cos(2y/b)
Complexity–NET: (www.complexitynet.eu). The authors gratefully and acknowledge support by 0the UK and Dutch funding agencies involved in Complexity–NET: the EPSRC and the NWO. The authors −2bsin(x)sin(y/b) to2bcos(x)sin(y/b) institutes for kind hospitality. are also indebted their respective . q2 = 0 to thank Jan Barkmeijer, Theodoros Finally, the authors wish −2bsin(x)sin(2y/b) Economou, Chris 2bcos(x)sin(2y/b) Ferro, Ken Mylne, and David Stephenson for useful discussions and remarks. Note that the vectors q and q are orthogonal. 1
2
Acknowledgements. This research is part of the project “PREDEX: Edited by: S. Vannitsem Predictability of Extreme Weather Events”, funded by Complexity– Reviewed by: two anonymous referees NET: (www.complexitynet.eu). The authors gratefully acknowledge support by the UK and Dutch funding agencies involved in
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