Lorenz Lecture - Nonlinear Geophysics

Lorenz Lecture AGU Fall Meeting, December 14, 2016 Analytic theory

Analytic theory of wind-driven seas Historical tour Basics Statistical theory

Vladimir E. Zakharov University of Arizona, Tuscon, USA

Universality of wave growth The theory for wave forecasting Conclusions References

in collaboration with S.I. Badulin (P.P. Shirshov Institute of Oceanology, Moscow) V.V. Geogjaev (P.P. Shirshov Institute of Oceanology, Moscow) A.N. Pushkarev (P.N. Lebedev Institute of Physics, Moscow)

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Outlines Analytic theory

1

Sea wave forecasting history

2

Hamiltonian equations for water waves and the kinetic equation

Universality of wave growth

3

Statistical description of wind waves

The theory for wave forecasting

4

Universality of wind wave growth

5

The theory for wave forecasting

6

Conclusions

7

References

Historical tour Basics Statistical theory

Conclusions References

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Old is always gold Analytic theory

Historical tour

The Beaufort Scale is an empirical measure for describing wind speed based mainly on observed sea conditions. Its full name is the Beaufort Wind Force Scale.

Basics Statistical theory Universality of wave growth The theory for wave forecasting Conclusions References

Sir Francis Beaufort (1774-1857) 3 / 55

Ivan Ivazovsky. Brig Mercury attacked by two Turkish ships (1892). Beaufort force 6 Analytic theory

Historical tour Basics Statistical theory Universality of wave growth The theory for wave forecasting Conclusions References

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Ivan Ivazovsky. The ninth wave (1850) Beaufort force 12 Analytic theory

Historical tour Basics Statistical theory Universality of wave growth The theory for wave forecasting Conclusions References

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Weather and wave forecasting Analytic theory

Urbain Jean Joseph Le Verrier (1811-1877)

Robert FitzRoy (1805-1865)

Historical tour Basics Statistical theory Universality of wave growth The theory for wave forecasting Conclusions References

Meteo-France, 1854 Met Office, UK, 1854 Both are created just after the Great Storm 14 November 1854 (the Crimean War). Anti-Russian coalition lost 1500 men, 53 ships incl. 25 cargos, goods and ammunition for 60 mln. francs 6 / 55

The old is always new The World War II and the concept of significant wave height Analytic theory

Historical tour Basics Statistical theory Universality of wave growth The theory for wave forecasting Conclusions References

In particular, by studies for the Harald Ulrik Sverdrup Overlord operation Walter Heinrich Munk 1888-1957 1917 (1944)

‘The concept of ‘significant waves’ is essential for purpose of forecasting’ 7 / 55

The theory Analytic theory

Historical tour

η(r, t) − surface elevation

Basics Statistical theory Universality of wave growth The theory for wave forecasting Conclusions References

r = (x, y ) − horizontal coordinates divV = 0;

V = ∇Φ;

∆Φ = 0

∂η δH ∂Ψ δH = ; = − ; H = T + U − total energy ∂t δΨ ∂t δη Z Z 1 1 ∂Ψ 2 U= dS − kinetic η dr − potential; H = Ψ 2 2 ∂n

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Theory of weakly nonlinear waves Analytic theory

Historical tour Basics Statistical theory Universality of wave growth The theory for wave forecasting Conclusions

T =

1 2

Z

Z

η

dr r

(∇Φ)2 dz = −

Z

G (s, s 0 )Ψ(s)Ψ(s 0 )dsds 0

∞

G (s, s 0 ) = G (s 0 , s) – the Green function of the Dirichlet-Neuman problem Φ = Ψ|z=h ;

References

H = H0 + H1 + H2 + . . . ;

∂Ψ → 0, ∂z

z →∞

µ∼ = kη – average steepness

µ2 + µ 3 + µ4

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Hamiltonian equations and normal variables Analytic theory

We use non-symmetric Fourier transform Z Z η(r, t) = η(k) exp(ikr)dk; Ψ(r, t) = Ψ(k) exp(ikr)dk

Historical tour Basics Statistical theory Universality of wave growth The theory for wave forecasting

Normal variables   1/2  g 1/2 ωk ∗ ∗ (ak + a−k ); Ψk = (ak − a−k ) ηk = 2g 2ωk Z p H = ωk ak ak∗ dk + Hint ; ωk = g |k| − dispersion law

Conclusions References

∂ak δ H˜ = i ∗; ∂t δak

1 H˜ = 2 H 4π

hak ak∗0 i = Nk δ(k − k0 );

εk = ωk Nk 10 / 55

The most important nonlinear interactions Analytic theory

Historical tour Basics

4-wave resonances 

ω0 + ω1 = ω4 + ω3 k0 + k1 = k4 + k3

Statistical theory Universality of wave growth The theory for wave forecasting Conclusions References

Phillips’ curve 

2ω0 = ω4 + ω3 2k0 = k4 + k3 11 / 55

Owen Martin Phillips (1930–2010) Analytic theory

Historical tour Basics Statistical theory Universality of wave growth The theory for wave forecasting Conclusions References

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Canonical transformation – eliminating three-wave interactions – the effective Hamiltonian Analytic theory

Canonical transformation of normal variables Historical tour Basics

ak → bk

Statistical theory Universality of wave growth The theory for wave forecasting Conclusions References

Z H⇒

ωk bk bk∗ +

1 2

Z

Tkk1 k2 k3 bk∗ bk∗1 bk2 bk3 δk+k1 −k2 −k3 dkk1 k2 k3

T (εk, εk1 , εk2 , εk3 ) = ε3 T (k, k1 , k2 , k3 )

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Matrix element T (we put g = 1 here) Analytic theory

Historical tour Basics Statistical theory Universality of wave growth The theory for wave forecasting Conclusions References

n 1 1 T˜k1 k2 k3 k4 = − 4 (k1 k2 k3 k4 )1/4
  1 2 k1+2 − (ω1 + ω2 )4 (k1 k2 − k1 k2 ) + (k3 k4 − k3 k4 ) + 2
  1 2 − k1−3 − (ω1 − ω3 )4 (k1 k3 + k1 k3 ) + (k2 k4 + k2 k4 ) 2
  1 2 k1−4 − (ω1 − ω4 )4 (k1 k4 + k1 k4 ) + (k2 k3 + k2 k3 ) − 2   4(ω1 + ω2 )2 + − 1 (k1 k2 − k1 k2 )(k3 k4 − k3 k4 ) k1+2 − (ω1 + ω2 )2   4(ω1 − ω3 )2 + − 1 (k1 k3 + k1 k3 )(k2 k4 + k2 k4 ) k1−3 − (ω1 − ω3 )2 )   4(ω1 − ω4 )2 + − 1 (k1 k4 + k1 k4 )(k2 k3 + k2 k3 ) k1−4 − (ω1 − ω4 )2

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We need in statistical description Analytic theory

Historical tour Basics Statistical theory Universality of wave growth The theory for wave forecasting Conclusions References

hbk bk∗0 i = Nk δk−k0 ; – Nk – spectrum of wave energy satisfying the Hasselmann equation dNk dt Snl

= Snl = πg

2

Z

|Tkk1 k2 k3 |2 dkdk1 dk2 dk3

k1 ,k2 ,k3

(Nk Nk1 Nk2 + Nk Nk1 Nk4 − Nk Nk2 Nk3 − Nk1 Nk2 Nk3 ) × δ(k + k1 − k2 − k3 )δ(ω + ω1 − ω2 − ω3 ) p ω = gk − dispersion law Energy spectrum (real energy spectrum is ρw g εk ) – εk = ωk Nk Spectrum of surface elevation – Ik = h|ηk |2 i = 21 ωk (Nk + N−k ) 15 / 55

Klaus Hasselmann (JFM, 1962) Analytic theory

Historical tour Basics Statistical theory Universality of wave growth The theory for wave forecasting Conclusions References

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What is the energy spectrum? Analytic theory

ε(k) = ωk Nk Historical tour Basics Statistical theory Universality of wave growth The theory for wave forecasting

ε(k)dk =

2ω 3 ε(ω, θ)dωdθ g2

Z 2π 1 ε(ω) = ε(ω, θ)dθ- omnidirectional spectrum 2π 0 Z Z ε(ω)dω = ε(f )df = hη 2 i = σ 2 - variance

Conclusions References

With a good accuracy the amplitude distribution is gaussian P(η) = √

1 η2 exp(− 2 ) 2σ 2πσ 17 / 55

the Rayleigh distribution

Analytic theory

Historical tour

P(H) =

Basics Statistical theory Universality of wave growth The theory for wave forecasting

√ 2πσ ≈ 2.5σ √ = hH 2 i1/2 = 2 2σ ≈ 2.83σ hHi =

Hrms

Hs – significant height;

Conclusions References

H H2 exp(− ) 4σ 2 8σ 2

Hs = 4σ

could be 30 m or even more H1/10 ≈ 5.09σ Hrogue > 2.1Hs > 8.4σ 2

In a strong storm Hs ≈ 0.22 Ug < 14 m – non-gaussian PDF

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From the concept of significant wave height to the spectral theory of wind seas (Gelci et al. 1957) Analytic theory

Historical tour Basics Statistical theory Universality of wave growth The theory for wave forecasting Conclusions References

Wave spectra - decomposition in wave scales (frequency or wavenumbers) Z ∞ E= E (f )df 0 19 / 55

More ideas on universal shaping (similarity) of wave spectra Analytic theory

Historical tour

The spectra do not depend (depend slightly) on conditions of wave spectra evolution

Basics Statistical theory Universality of wave growth The theory for wave forecasting Conclusions References

Donelan et al. 1985 20 / 55

More ideas on spectral shapes in log-axes Analytic theory

O. Phillips, 1985

P. Liu, 1989

Historical tour Basics Statistical theory Universality of wave growth The theory for wave forecasting Conclusions References

Hwang & Wang, 2004 21 / 55

Donelan & Kahma, 1986 Analytic theory

Historical tour Basics Statistical theory Universality of wave growth The theory for wave forecasting Conclusions References

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Spectral tail Analytic theory

Historical tour Basics Statistical theory Universality of wave growth The theory for wave forecasting Conclusions References

Numerous experiments performed in wave tanks as well as field observations show that the tail of the omnidirectional wave energy spectrum E (ω) '

αgU ; ω4

α – dimensionless constant

Resio, Long & Vincent (2004) offer more accurate formula E (ω) '

α4 g (U 2 Cp )1/3 ; ω4

F (k) ' βk −5/2

Cp – group velocity of the spectral maximum α4 ≈ 5.5 · 10−3 23 / 55

Wave spectra scaling by energy flux Analytic theory

Historical tour Basics Statistical theory Universality of wave growth The theory for wave forecasting Conclusions References

1/3

For F (k) = βk −5/2 ⇔ ε = Ck Pω4

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Wave growth and wind speed scaling Analytic theory

Following Kitaigorodskii (1962) we introduce dimensionless quantities

Historical tour Basics Statistical theory Universality of wave growth The theory for wave forecasting Conclusions References

2 χ = xg /U10 4 ε˜ = σ 2 g 2 /U10

ω ˜ = ωU10 /g

ωp – frequency of the spectral peak U10 – wind speed at the ‘standard animometer’ height 10 meters

Experiments show that in the fetch-limited conditions (spectrum does not depend on time) ε˜ and ω ˜ are power-like functions ε˜ = ε˜0 χp ω ˜=ω ˜ 0 χ−q

0.7 < p < 1.1;

0.68 < 107 ε˜0 < 18.6

0.23 < q < 0.33;

10.4 < ω ˜ 0 < 22.6 25 / 55

I. R. Young, 1999 Analytic theory

Historical tour Basics Statistical theory Universality of wave growth The theory for wave forecasting Conclusions References

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Exponents of fetch-limited growth in experiments Analytic theory

Experiment

pχ

qχ

zχ

Black Sea

0.89

0.275

0.010

Walsh et al. 1989, US coast

1.

0.29

0.033

Historical tour

Kahma & Calkoen, 1992, unstable

0.94

0.28

0.027

Basics

Kahma & Calkoen 1992, stable

0.76

0.24

0.040

Kahma & Pettersson 1994 Davidan 1980

0.93 1.

0.28 0.28

0.020 0.067

JONSWAP by Phillips, 1977

1.

0.25

0.167

Kahma& Calkoen, 1992, composite

0.9

0.27

0.033

Kahma 1981, 1986, rapid growth

1.

0.33

-0.100

The theory for wave forecasting

Kahma 1986, average growth

1.0

0.33

-0.100

Donelan et al, 1992, St. Claire

1.0

0.33

0.023

Conclusions

Ross, 1978 unstable, Michigan

1.1

0.27

0.167

Liu & Ross, 1980, stratification

1.1

0.27

0.167

JONSWAP Hasselmann et al.,l973

1.

0.33

-0.010

Mitsuyasu, 1971

1.008

0.33

-0.095

ZRP numerics

1

0.3

0

Statistical theory Universality of wave growth

References

2pχ − 10qχ − “magic relation” mismatch 3 Adopted from Badulin, Babanin, Zakharov, Resio 2007 zχ =

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‘Magic links’ in sea experiments Analytic theory

Historical tour Basics Statistical theory Universality of wave growth The theory for wave forecasting Conclusions References

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Universality of wind-wave growth Analytic theory

Historical tour Basics

‘Magic links’ and spectral features of wind-waves can be summarized in a remarkably simple wind-free algebraic relationship

Statistical theory Universality of wave growth

µ4ν = α03

The theory for wave forecasting Conclusions References

µ= ν = ωp t α0 ≈

hakp i

– wave steepness

(2kp x) – number of waves 0.7

– a universal constant 29 / 55

Our wind-free invariant in sea experiments Analytic theory

Historical tour Basics Statistical theory Universality of wave growth The theory for wave forecasting Conclusions References

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Theoretical explanation Analytic theory

Go back to the resonant conditions k1 + k2 = k3 + k4

Historical tour

ω1 + ω2 = ω3 + ω4

Basics Statistical theory

Long-short interaction, asymptotics If k1  k2 , k3  k4 then k2 ≈ k4

Universality of wave growth

|k1 | ≈ |k3 |

The theory for wave forecasting Conclusions References 0.4 0.3 0.2 0.1 0 -0.1 -0.2 -0.3 -0.4 -1.5

-1

-0.5

0

0.5

1

1.5

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Kernel T asymptotics Analytic theory

Historical tour Basics Statistical theory Universality of wave growth The theory for wave forecasting Conclusions References

1 Tk,k1 ,k2 ,k3 = − |k|2 |k2 |Tθ1 θ2 2 Tθ1 θ2 = (cos θ1 + cos θ3 ) cos(θ1 − θ3 ) θ1 and θ2 are the angles between the small vectors and the base vector This small parameter expansion has two first orders cancelled. As a result, the long-short interactions are weak

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Analytic theory

Historical tour Basics Statistical theory Universality of wave growth The theory for wave forecasting Conclusions References

Let us consider the stationary Hasselmann equation Snl = 0 The ‘naive’ solutions T Nk = ωk + µ are not solutions because of integral divergence.

The wind-driven sea is very far from the thermodynamic equilibrium. Similar to turbulence in incompressible fluid in the isotropic case the Hasselmann equation has two conservation laws: energy ε and wave action N Z Z ε = ωk Nk dk; N = Nk dk 33 / 55

Isotropic power-like solutions Analytic theory

Let N = k −x . Then 3

Statistical theory Universality of wave growth The theory for wave forecasting

F (x) is defined for F → F →

5 2