David Henry University College Cork, Irelandrivanov/Workshop/Workshop Talks/Henry.pdf ·...

Introduction Modelling fluid dynamics Equatorial water waves Local existence theory Dispersion relations

Equatorial wind waves

David HenryUniversity College Cork, Ireland

Workshop on Mathematics of Nonlinear PhenomenaDublin Institute of Technology, 8th May, 2014


“The sea is emotion incarnate. It loves, hates, and weeps. Itdefies all attempts to capture it with words and rejects allshackles. No matter what you say about it, there is always thatwhich you can’t.” Paolini


Introduction

In this talk I will outline some recent results which haveappeared in the paper:


Introduction

In this talk I will outline some recent results which haveappeared in the paper:

DH and Anca-Voichita Matioc, On the existence of equatorialwind waves, Nonlinear Anal. 101 (2014), 113–123.


Introduction

Although waves in water are a commonplace, readilyobserved phenomenon, a rigorous mathematical treatmentof fluid motion was first established as late as the 18th

century.


Introduction


century.

It was Daniel Bernouilli who introduced the termhydrodynamics to comprise hydrostatics and hydraulics.


Introduction


century.


The foundations of the theory of modern hydrodynamicswere laid by Bernouilli, Euler, Lagrange and d’Alembert,among others, during this time.


Introduction


century.


The foundations of the theory of modern hydrodynamicswere laid by Bernouilli, Euler, Lagrange and d’Alembert,among others, during this time.

Further major advances in the theory were achieved in the19th century, notably by Navier and George Gabriel Stokes.


Overview

Mathematics in fluid dynamics

Hydrodynamics is very rich branch of mathematics which lieson the interface of pure and applied maths.


Overview


Hydrodynamics is very rich branch of mathematics which lieson the interface of pure and applied maths. The importance ofhydrodynamics in contemporary mathematical research isoutlined nicely by the following quote from V. I. Arnold:


Overview


Hydrodynamics is very rich branch of mathematics which lieson the interface of pure and applied maths. The importance ofhydrodynamics in contemporary mathematical research isoutlined nicely by the following quote from V. I. Arnold:

“Hydrodynamics is one of those fundamental areas whereprogress at any moment can be regarded as a standard tomeasure the real success of mathematical science”.


Overview

Nonlinear governing equations

The governing equations for water waves, of all varioustypes, take the form of partial differential equations (PDEs).


Overview



The full equations are nonlinear.


Overview




Additionally, we must solve a free boundary problem, sincethe wave surface is not known a priori.


Overview




Additionally, we must solve a free boundary problem, sincethe wave surface is not known a priori.

Determining the wave surface is part of the overall problemitself.


Overview

Nonlinear governing equations (perfect fluid)

Geophysical equations (includes effects of Earth’s rotation)

R = 6378km,Ω = 73 · 10−6rad/s, φ the latitude:

ut + uux + vuy + wuz + 2Ωw cosφ− 2Ωv sinφ = −1ρ

Px ,

vt + uvx + vvy + wvz + 2Ωu sinφ = −1ρ

Py ,

wt + uwx + vwy + wwz − 2Ωu cosφ = −1ρ

Pz − g,

Here x , y are the horizontal, and z the vertical, variables,(u, v ,w) the velocity field, P the pressure function.


Overview

Nonlinear governing equations (perfect fluid)

Geophysical equations (β−plane: latitudes < 5)

sinφ ≈ φ, cosφ ≈ 1 (since φ small):

ut + uux + vuy + wuz + 2Ωw − βyv = −1ρ

Px ,

vt + uvx + vvy + wvz + βyu = −1ρ

Py ,

wt + uwx + vwy + wwz − 2Ωu = −1ρ

Pz − g,

Here β = 2Ω/R = 2.28 · 10−11m−1s−1.


Overview

Due to the nonlinearity of the equations, and the unknownfree boundary, the governing equations are mathematicallyhighly intractable to exact analysis.


Overview

Due to the nonlinearity of the equations, and the unknownfree boundary, the governing equations are mathematicallyhighly intractable to exact analysis.

There are only a handful of known explicit solutions, andanalysing the equations requires powerful techniques frompure and applied maths.


Overview

Importance of nonlinearity

The crests are much narrower than the troughs are wide: theprofile is obviously not sinusoidal. Therefore linear theory is notsufficient for modelling this scenario. Note the surfers for scale!


Geophysical water waves


Geophysical fluid dynamics is the study of large-scalephysical phenomena where the effect of the Earth’srotation plays a significant role and therefore must be takeninto account through the presence of Coriolis forces in thegoverning equations.





Geophysical processes which occur in the equatorialregion are of particular interest for a number of reasons.





Geophysical processes which occur in the equatorialregion are of particular interest for a number of reasons.

Physically, the equator has the remarkable property ofacting as a natural wave guide, whereby equatoriallytrapped zonal waves decay exponentially away from theequator in the oceans.



Allied to this, large-scale currents and wave-currentinteractions play a major role in the geophysical dynamicsof the equatorial region.



Allied to this, large-scale currents and wave-currentinteractions play a major role in the geophysical dynamicsof the equatorial region.

Among the most spectacular in scale is the EquatorialUndercurrent (EUC) which extends practically throughoutthe Pacific Ocean, 13000km, and which exists in arelatively shallow layer tens of metres beneath the surface.


El Niño and the Equatorial Undercurrent

Aside: El Niño and the Equatorial Undercurrent (EUC)

“El Niño”— Spanish for “The Christ Child”, and thus namedby early fishermen— is an event associated with theappearance around the Christmas season of an oceananomaly.





El Niño is a band of anomalously warm ocean watertemperatures that periodically develops off the westerncoast of South America and can cause climatic changesacross the Pacific Ocean.






El Niño is a reversal of the normal situation in the PacificOcean:






El Niño is a reversal of the normal situation in the PacificOcean: usually, the warm surface water is blownwestwards by prevailing winds, and deeper (thereforecolder) water is forced upwards to replace it.



However, every now and then, the surface water sloshesback across the ocean, bringing warm water along theeastern coasts of the pacific.




The exact mechanisms by which this reversal happens iseven now a mystery, however in recent decades a numberof discoveries have clarified aspects of this process.





It is now possible to follow, as they happen, the majorchanges in the circulation of the tropical Pacific Ocean thataccompany the alternate warming and cooling of thesurface waters of the eastern equatorial Pacific associatedwith El Niño and La Niña.





It is now possible to follow, as they happen, the majorchanges in the circulation of the tropical Pacific Ocean thataccompany the alternate warming and cooling of thesurface waters of the eastern equatorial Pacific associatedwith El Niño and La Niña.

Satellite-borne radiometers and altimeters measure oceantemperature and sea level height almost in real time,providing information on slow (interannual) changes in theocean thermal structure as well as frequent glimpses ofswift wave propagation.



The existence of the Pacific Equatorial Undercurrent (EUC)was discovered, fortuitously, in the 1950s, and it is nowknown to play a major role in equatorial dynamics.




The Pacific Equatorial Undercurrent is an eastward-flowingsubsurface current that extends the length of the equator inthe Pacific Ocean.




The Pacific Equatorial Undercurrent is an eastward-flowingsubsurface current that extends the length of the equator inthe Pacific Ocean.

The surface currents flow west. There is reversal pointabout 40m down, where the water starts to flow east. Thecurrent goes down to about 400m.



The thinness, symmetry, and length of the EUC are trulyremarkable: the Pacific EUC is confined to a shallowsurface layer centered on the Equator and less than 200mdeep, it is typically about 300km in width and symmetricabout the Equator.




The total flow is up to around 30, 000, 000 cubic meters persecond. The top speed is around 1.5m/s (this is abouttwice as fast as the westerly surface current) and extendsnearly across the whole length (about 13, 000km) of theocean basin.




The total flow is up to around 30, 000, 000 cubic meters persecond. The top speed is around 1.5m/s (this is abouttwice as fast as the westerly surface current) and extendsnearly across the whole length (about 13, 000km) of theocean basin.

It has 1000 times the volume of Mississippi River.



Mathematical modelling of equatorial wind waves andwave current interactions

Equatorial waves are trapped, since the equator acts as anatural wave guide.





This enables us to employ the assumption oftwo-dimensionality of the fluid motion.






Furthermore, we seek periodic, steady waves, propagatingwestwards. Therefore the wave phase-speed c < 0.







Equatorial waves tend to flow westwards due to theprevailing wind direction.







Equatorial waves tend to flow westwards due to theprevailing wind direction.

Without loss of generality we choose the period to be 2π.



The reversal of the fluid velocity beneath the surface profileis then captured by allowing for a certain (negative)vorticity distribution in the water flow.



The reversal of the fluid velocity beneath the surface profileis then captured by allowing for a certain (negative)vorticity distribution in the water flow.

The presence of vorticity in a flow is necessary to modelwave-current interactions.



Two-dimensional, steady periodic motion of fluids

For ocean waves, and in particular swell generated byprevailing ocean winds, the motion appears regular.




Geophysical equations in the f−plane (equatorial region)

ut + uux + wuz + 2Ωw = −Px/ρ,

wt + uwx + wwz − 2Ωu = −Pz/ρ− g,

ux + wz = 0,

P = Patm on z = η(t , x , y),

w = ηt + uηx on z = η(t , x , y),

w = 0 on z = −d .




Stream-function formulation: ψz = u − c, ψx = −w

∆ψ = γ(ψ) in − d < z < η(x),

|∇ψ|2 + 2(g − 2Ωc)z = Q on z = η(x),

ψ = 0 on z = η(x),

ψ = m on z = −d ,

The constants of motion are the total head Q, and mass-flux m.The function γ = uz − wx is the vorticity of the flow: we admitgeneral vorticity distributions.




Height-function formulation: h = z + d .

(1 + h2q)hpp − 2hphqhpq + h2

phqq + γ(p)h3p = 0 in R,

1 + h2q + [2(g − 2Ωc)(h − d)− Q]h2

p = 0 on p = 0,

h = 0 on p = m.(1)

For c < u, the Dubreil-Jacotin semi-hodgraph transformation(q, p) = (x , ψ) maps the fluid domain onto (fixed) rectangle R.


Existence of small-amplitude waves

Crandall-Rabinowitz Theorem

Theorem

Let X ,Y be Banach spaces and let F ∈ Ck (R× X ,Y ) withk ≥ 2 satisfy:

(a) F(γ, 0) = 0 for all γ ∈ R;(b) The Fréchet derivative ∂xF(γ∗, 0) is a Fredholm operator

of index zero with a one-dimensional kernel:

Ker(∂xF(γ∗, 0)) = sx0 : s ∈ R, 0 6= x0 ∈ X;

(c) The tranversality condition holds:

∂γxF(γ∗, 0)[(1, x0)] 6∈ Im(∂xF(γ∗, 0)).



Theorem

Then γ∗ is a bifurcation point in the sense that there existsǫ0 > 0 and a branch of solutions

(γ, x) = (Γ(s), sχ(s)) : s ∈ R, |s| < ǫ0 ⊂ R× X ,

with F(γ, x) = 0, Γ(0) = γ∗, χ(0) = x0, and the maps

s 7→ Γ(s) ∈ R, s 7→ sχ(s) ∈ X ,

are of class Ck−1 on (−ǫ0, ǫ0). Furthermore there exists anopen set U0 ⊂ R× X with (γ∗, 0) ∈ U0 and

(γ, x) ∈ U0 : F(γ, x) = 0, x 6= 0 = (Γ(s), sχ(s)) : 0 < |s| < ǫ0.



Laminar flow solutions to the height-function formulation exist,and are given by

H(p;λ) =∫ p

m

ds√

λ+2Γ(s)




H(p;λ) =∫ p

m

ds√

λ+2Γ(s)

where the parameter λ > 0 is defined by√λ = u(0;λ)− c,




H(p;λ) =∫ p

m

ds√

λ+2Γ(s)

where the parameter λ > 0 is defined by√λ = u(0;λ)− c,

and

Γ(p) =∫ p

0γ(s)ds.



Then, for h(q, p) = H(p;λ) + f (q, p) the height functiongoverning equations can be expressed in operator form

F(λ, f ) = 0



Then, for h(q, p) = H(p;λ) + f (q, p) the height functiongoverning equations can be expressed in operator form

F(λ, f ) = 0

with f ∈ X = h ∈ C2,αper (R), h = 0 on p = m.



Here F = (F1,F2) : X × (−2Γmin,∞) → Y



Here F = (F1,F2) : X × (−2Γmin,∞) → Y , forY := Y1 × Y2 = C1,α

per (R)× C2,αper (S)




per (R)× C2,αper (S), is given by

F1(f , λ) =(

1 + f 2q

)

(Hpp + fpp)− 2fqfqp(Hp + fp)

+ fqq(Hp + fp)2 + γ(p)(Hp + fp)3,




per (R)× C2,αper (S), is given by

F1(f , λ) =(

1 + f 2q

)

(Hpp + fpp)− 2fqfqp(Hp + fp)

+ fqq(Hp + fp)2 + γ(p)(Hp + fp)3,

F2(f , λ) =(

1 + f 2q + [2(g − 2Ωc)(H + f − d)− Q](Hp + fp)2

)

T,

where the subscript T denotes the trace operator.



We haveF(λ, 0) = 0 for all λ.



We haveF(λ, 0) = 0 for all λ.

The laminar flows play the role of trivial solutions in theCrandall-Rabinowitz Theorem.



It can be shown that when the vorticity satisfies

g > 2Ωc +

√2

3γ

32∞|p1|

12 +

2√

25

γ12∞|p1|

32 ,




g > 2Ωc +

√2

3γ

32∞|p1|

12 +

2√

25

γ12∞|p1|

32 ,

with γ∞ = ‖γ‖C([m,0]) and p1 = minp ∈ [m, 0] : Γ(p) = Γmin,




g > 2Ωc +

√2

3γ

32∞|p1|

12 +

2√

25

γ12∞|p1|

32 ,

with γ∞ = ‖γ‖C([m,0]) and p1 = minp ∈ [m, 0] : Γ(p) = Γmin,there is a critical point of the bifurcation parameter at which theoperator, and its Fréchet derivatives, fulfill the criteria of theCrandall-Rabinowitz local bifurcation theorem.



An immediate consequence of this result is that localbifurcation always occurs when the vorticity γ ≤ 0, for inthis case p1 = 0.



An immediate consequence of this result is that localbifurcation always occurs when the vorticity γ ≤ 0, for inthis case p1 = 0.

Furthermore, since c < 0 the above condition is (slightly!)less-restrictive than the analogous condition which holdsfor water waves without Coriolis effects.



The technical aspects of this analysis encompass the spectraltheory of operators, an investigation of the groundstate forweighted Sturm-Liouville problems, and Schauder estimatestogether with general elliptic regularity theory for nonlinear,uniformly elliptic equations which have nonlinear, obliqueboundary conditions.


Dispersion relations

A very useful by-product of our analysis, when analysingthe Fréchet derivatives of our operator, is the determinationof the dispersion relation for the flow.




If ud = u(−d) is the constant horizontal speed at thebottom of the fluid layer




If ud = u(−d) is the constant horizontal speed at thebottom of the fluid layer, then

c = ud + γd − (γ + 2Ω) tanh d2

(2)

−

√

(

(γ + 2Ω) tanh d2

)2

+ (g − 2Ω(ud + γd)) tanh d , (3)



For waves of general wavelength L, performing the followingscaling of variables

(x , z, t , g, γ, ω, η, u,w ,P, c) 7→ (4)

(κx , κz, κt , κ−1g, κ−1γ, κ−1ω, κη, u,w ,P, c) (5)



For waves of general wavelength L, performing the followingscaling of variables

(x , z, t , g, γ, ω, η, u,w ,P, c) 7→ (4)

(κx , κz, κt , κ−1g, κ−1γ, κ−1ω, κη, u,w ,P, c) (5)

where κ = 2πL is the wavenumber, we end up with a

2π−periodic system in the new variables (except g, ω, γ arereplaced by κ−1g, κ−1ω, κ−1γ).



For waves of general wavelength, the dispersion relation thenbecomes:



For waves of general wavelength, the dispersion relation thenbecomes:

c = ud + γd − (γ + 2Ω) tanh(κd)2κ

−

√

(

(γ + 2Ω) tanh(κd)2κ

)2

+ (g − 2Ω(ud + γd))tanh(κd)

κ.

David Henry University College Cork, Irelandrivanov/Workshop/Workshop Talks/Henry.pdf ·...

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