MSc Thesis Defense: - PowerPoint Presentation

Slide1

MSc Thesis Defense: Impact of static sea surface topography variations on ocean surface waves

Student:

YING, Yik Keung (EM-COSSE)

Supervisors:

Prof C.

Vuik

(Delft), Prof L.R. Maas (NIOZ)Slide2

Content Page1. Background - Gravitational Fields and Surface Waves

2. Waves in Shallow Water

a. Standard Shallow Water Model

b. Adapted Shallow Water Model

i

. One-Dimensional Waves

ii. Two-Dimensional Waves

3.

Airy’s

Linear Wave Theory

a. Standard

Airy’s

Linear Wave Theory

b. Generalised

Airy’s

Linear Wave Theory

4. Discussions

5. ConclusionsSlide3

Content Page1. Background - Gravitational Fields and Surface Waves

2. Waves in Shallow Water

a. Standard Shallow Water Model

b. Adapted Shallow Water Model

i

. One-Dimensional Waves

ii. Two-Dimensional Waves

3.

Airy’s

Linear Wave Theory

a. Standard

Airy’s

Linear Wave Theory

b. Generalised

Airy’s

Linear Wave Theory

4. Discussions

5. ConclusionsSlide4

Background – Gravitational Field, Theory

Poisson’s Equation (or Laplace Equation)

Equipotential Surface

collection of points

(

x,y,z

)

such that

Φ

a

: Attractive Potentialρ: Density distributionG: Universal Constant

Φ

:

Conservative Potential

Φ

0

: ConstantSlide5

Background – Mean-Sea LevelShape of Earth:

The Mean-Sea Level as an Equipotential SurfaceSlide6

Background – Gravitational Field on Earth

The gravity is never uniform!

1

milligal

= 1 cm/s^2Slide7

Background – Ocean Surface Waves

Example: SwellsSlide8

Background – Ocean Surface Waves

Example: Tsunami wavesSlide9

Content Page

1. Background - Gravitational Fields and Surface Fluid Waves

2. Waves in Shallow Water

a. Standard Shallow Water Model

b. Adapted Shallow Water Model

i

. One-Dimensional Waves

ii. Two-Dimensional Waves

3. Airy’s

Linear Wave Theorya. Standard Airy’s Linear Wave Theoryb. Generalised Airy’s

Linear Wave Theory4. Further Discussions5. ConclusionsSlide10

Shallow Water Model: Standard

Fluid surface

z=0

Water

Depth

H

Wave Height 2a

Crest

Wavelength

Water amplitude a

z

x

: surface elevation;

: water depth;

: wave speed

: mass flux

Governing equations for shallow water waves

Shallowness

:

Aspect ratio

Small amplitude

 Slide11

b. Implication

:

Horizontal pressure gradient is determined by surface elevation

Shallow Water Model: Standard

Assumptions:

1. Ideal fluid

2. Uniform Gravity

3. Shallowness:

a. 2.5D formalism

b. Hydrostatic approximation

: scale of vertical velocity;

: scale of horizontal velocity;

: aspect ratio of length scales of motion;

: pressure;

: surface elevation;

:

atmo

. pressure (const.)

: gravity (const.)

: density of fluid (const.)

a. Implication

:

is negligible compare to

in shallow water

 Slide12

Shallow Water Model: Adapted

Assumptions:

1. Ideal fluid

2.

Conservative Gravity

3. Shallowness:

a. 2.5D formalism

b. Hydrostatic approximation

: scale of vertical velocity;

: scale of horizontal velocity;

: aspect ratio of motion;

Question:

How to deal with the hydrostatic approximation when gravity is non-uniform?Slide13

Shallow Water Model: AdaptedHow does the hydrostatic condition work?

Recast the vertical coordinates

via the conservative potential

Z-Transformation on vertical coordinates

Potential Difference

Ψ

with MSL

Z-coordinate function

: P.D. with MSL;

: Potential function;

: Potential at MSL;

: Reference gravity (const.)

: hydrostatic pressure

: Potential function

: density of fluid (const.)

 Slide14

Shallow Water Waves: Adapted ModelVisualisation of the (x, Z) coordinates

Horizontal Coordinate, x

Equipotential Lines, Z

0

Re-definition of ‘depth’!Slide15

Shallow Water Model: AdaptedHydrostatic condition in

(x, y, Z)

coordinates

After hydrostatic approximation in

(x, y, Z)

coordinates…

Horizontal pressure gradient:

: hydrostatic pressure

: Potential function

: density of fluid (const.)

: Reference gravity (const.)

Analogous to classical case

: dynamic pressure

: coordinates function

Horizontal gradientSlide16

Shallow Water Waves: Adapted ModelTransformed governing equations

Additional assumptions:

1.

Shallowness

: Kills nonlinear Jacobian term in momentum

2.

During depth-averaging

: Gravity variation only at surface

Continuity

Momentum

: horizontal velocities;

: vertical velocity

: dynamic pressure;

: inverse coordinate function for z

Jacobian terms after coordinates transformation

Scale

:

aspect ratio of motion length scales

 Slide17

Shallow Water Waves: Adapted ModelDepth-averaging + Zero Normal Flow B.C.

Depth Averaging

Continuity

Momentum

Depth-averaged Continuity

Depth-averaged

Momentum

: Surface elevation;

: Hydrostatic depth;

: Horizontal velocity

Difference with standard shallow water model:

Adapting term in depth-averaged continuity equationSlide18

Shallow Water Waves: Adapted ModelConsider a small perturbation to quiescent fluid

Adapted Shallow Water Wave Equations

1. Wave speed:

2. Adapting term:

Difference with standard shallow water waves:

Adapting term in perturbed depth-averaged continuity equation

: surface elevation;

: water depth;

: wave speed

: mass flux

 Slide19

Adapted Shallow Water Waves:One-Dimensional

One-Dimensional Wave Equations

Consider the time-harmonic ansatz:

Recasting variables:

: surface elevation;

: water depth;

: wave speed

: mass flux

: redefined variable

: angular speed;

: amplitude field

 Slide20

Adapted Shallow Water Waves:One-Dimensional Diagnostic Formalism

Gives:

with diagnostic variables: ‘Kinetic Energy’ and ‘Potential’

Not physical quantities!

Solution: WKBJ-Approximation

: reference wavenumbers (const.)

Mild-slope: V(x) << E(x)Slide21

Adapted Shallow Water Waves:One-Dimensional Diagnostic Formalism

Solution by WKBJ-Approximation

Amplitude field:

Dependent on

g

z

;

Different from standard case

Wavenumber

fieldwhere

Time-harmonic ansatz

Short concluding remark:

Gravity: affect both amplitude and wavenumber of waves!Slide22

Adapted Shallow Water Waves:Numerical Solutions (1A)

Target: Validation of the WKBJ-Approximation

Test case: Hypothetical – Exponential Gravity Perturbation, blown-up

Left:

Hypothetical waves,

Right:

Hypothetical waves,

 Slide23

Adapted Shallow Water Waves:Numerical Solutions (1B)

Target: Validation of the WKBJ-Approximation

Test case: Hypothetical – Gaussian Gravity Perturbation, blown-up

Left:

Hypothetical waves,

Right:

Hypothetical waves,

 Slide24

Adapted Shallow Water Waves:Numerical Solutions (2A)

Target: Waves on the Ocean

Test case: Physical – Gaussian Gravity Perturbation, Tidal

Left:

Tidal waves,

,

Instant. diff.

,

Right:

Tidal waves,

,

Instant. diff.

,

 Slide25

Adapted Shallow Water Waves:Numerical Solutions (2B)

Target: Waves on the Ocean

Test case: Physical – Gaussian Gravity Perturbation, Tsunami

Left:

Tsunami waves,

,

Instant. diff.

,

Right:

Tidal waves,

,

Instant. diff.

 Slide26

Adapted Shallow Water Waves:Numerical Solutions (3)

Target: Waves on the Ocean

Test case: Physical – Gaussian Gravity & MSL Perturbation

Left:

Tsunami waves,

,

Instant. diff.

,

Right:

Tidal waves,

,

Instant. diff.

 Slide27

Concluding RemarksGravity -> Wave amplitude and wavenumber field

In the actual ocean:

Effect of gravity variation

Unlikely measurable

Effects of induced MSL variation

Maybe measurable?Slide28

Adapted Shallow Water Waves:Two-Dimensional

Two-Dimensional Wave Equations

Diagnostic formalism failed...

: surface elevation;

: water depth;

: wave speed

: mass flux

 Slide29

Adapted Shallow Water Waves:Numerical Solutions (7)

Target: Find out the difference between Gravity and Depth perturbation

Test case: Hypothetical – Identical Size Gaussian Gravity Perturbation vs MSL Perturbation

Positive perturbation

Negative perturbation

Only MSL perturbation

Gravity:

Change wave amplitudes!

= Shoaling

Only gravity perturbationSlide30

Adapted Shallow Water Waves:Numerical Solutions (4A)

Target: Waves on the Ocean

Test case: Physical – Gaussian Gravity Perturbation, Tidal waves

Positive perturbation

Negative perturbation

Spherical scattered waves

No perturbationSlide31

Adapted Shallow Water Waves:Numerical Solutions (4B)

Target: Waves on the Ocean

Test case: Physical – Gaussian Gravity Perturbation, Tsunami waves

Positive perturbation

Negative perturbation

Plane-waves like scattered waves

No perturbationSlide32

Adapted Shallow Water Waves:Numerical Solutions (5A)

Target: Waves on the Ocean

Test case: Physical – Gaussian Gravity & MSL Perturbations, Tidal waves

Positive perturbation

Negative perturbation

No perturbation

Minimal changes…Slide33

Adapted Shallow Water Waves:Numerical Solutions (6A)

Target: Waves on the Ocean,

filter out the effects of depth changes

Test case: Physical – Gaussian Gravity & MSL Perturbations, Tidal waves

Positive perturbation

Negative perturbation

Only MSL perturbation

Minimal changes againSlide34

Adapted Shallow Water Waves:Numerical Solutions (5B)

Target: Waves on the Ocean

Test case: Physical – Gaussian Gravity & MSL Perturbation, Tsunami waves

Positive perturbation

Negative perturbation

No perturbation

Maybe observable!?Slide35

Adapted Shallow Water Waves:Numerical Solutions (6B)

Target: Waves on the Ocean,

filter out the effects of depth changes

Test case: Physical – Gaussian Gravity & MSL Perturbation, Tsunami waves

Positive perturbation

Negative perturbation

Only MSL perturbation

Minimal changes againSlide36

Short Conclusions

Standard Model

Adapted Model

Wave speed

Adapting

term

Absent

Wave

scattering

By wave speed

By wave speed

Wave

Shoaling

By Depth

By

both Depth and Gravity

Standard Model

Adapted Model

Wave speed

Adapting

term

Absent

Wave

scattering

By wave speed

By wave speed

Wave

Shoaling

By Depth

By

both Depth and Gravity

Comparison between Standard and Adapted Shallow water waves

Inference 1:

Theoretically

distinguish waves scattered by Depth and Gravity

:

Spatially dependent

Inference 2:

In practice, it suffices to assume

but take into account of change of static water depth

due to gravity field

 Slide37

Content Page

1. Background - Gravitational Fields and Surface Fluid Waves

2. Waves in Shallow Water

a. Standard Shallow Water Model

b. Adapted Shallow Water Model

i

. One-Dimensional Waves

ii. Two-Dimensional Waves

3.

Airy’s Linear Wave Theorya. Standard Airy’s Linear Wave Theoryb. Generalised Airy’s Linear Wave Theory

4. Further Discussions5. ConclusionsSlide38

Airy’s

Linear Wave Theory: Standard

Continuity

Momentum

Boundary Conditions

Bottom:

Surface:

(

Linearised

)

Governing Equations:

Fluid surface

z=0

Water Depth h

Wave Height 2a

Crest

Wavelength

Water amplitude a

z

x

No assumption of shallowness!

: velocity potential;

: surface elevation;

: uniform gravity

 Slide39

Airy’s Linear Wave Theory: Standard

Analytical solution in uniformly deep ocean

Bottom:

(const.

)

Dispersion relation

Velocity Potential,

Surface Elevation,

: Surface elevation;

: Wave amplitude;

: Water depth (const.)

: Angular speed of waves

: Wavenumber

 Slide40

Airy’s Linear Theory: Generalised, 2D

Generalised

Airy’s

Linear Theory

Governing Equation:

Seemingly trivial?

Justified by

variational

principle!

Mean-sea level

: Potential Field function

Continuity

Momentum

Boundary Conditions

Bottom:

Surface:

: velocity potential;

Question:

Can we derive some analytical solutions from it?Slide41

Airy’s Linear Theory: Generalised, 2D

Yes we can.

Conservative force field provided a guide!

Recall:

Force potential

satisfies Laplace Equation in free space, i.e.

Consider 2D Laplace Equation

Harmonic conjugates => Conformal coordinates

Step 1:

Define vertical coordinate q

2

Step 2:

Apply Cauchy-Riemann condition to determine q

1

Step 3:

Rewrite Laplacian operator in (q

1

, q

2

)Slide42

Airy’s Linear Theory: Generalised, 2D

Visualisation of Coordinates (q

1

, q

2

)

Conformal Coordinates, q

1

Equipotential Lines, q

2

0Slide43

Airy’s Linear Theory: Generalised, 2DGoverning equations of linear waves after transformation:

Continuity

Momentum

Boundary Conditions

Bottom:

Surface:

: velocity potential;

same as classical!Slide44

Airy’s Linear Theory: Generalised, 2D

Analytical solution in ‘uniformly deep’ fluid

:

Standard result directly applicable

Velocity Potential,

Surface Elevation,

subject to dispersion relation:

: Surface elevation;

: Wave amplitude;

: Water depth;

: Angular speed of waves

: Wavenumber

 Slide45

Airy’s Linear Theory: Test CasesTest case 1: Fluid Waves around circle

Hydrostatic Fluid Interface

Solid Boundary

R

c

R

s

Equipotential Lines, q

2

Orthogonal

coordinates

Potential

Surface elevation

in polar coordinates

(r,

θ

)

: Surface elevation;

: Wave amplitude;

: MSL;

: Bottom boundary

: Wavenumber

: Reference gravity, const.

Remark: Periodic B.C.Slide46

Airy’s Linear Theory: Test CasesTest case 2: Fluid Waves in Decaying Perturbed Gravity Field

Potential

Orthogonal

coordinates

Example 2a: Linear waves at shorter wavelengths

:

Depth of source

;

: MSL

: Reference gravity, const.

: Reference const.

Example 2b: Linear waves at longer wavelengths

‘Perturbation’ coordinates to (

x,z

)Slide47

Airy’s Linear Theory: Test CasesConsistency with adapted shallow water model?

Step 1:

Increase wavelengths

(approach long-wave limit)

Step 2:

Compare amplitude field

Increasing wavelength

Adapted shallow water

Generalised

Airy’s

linear wavesSlide48

Content Page

1. Background - Gravitational Fields and Surface Fluid Waves

2. Waves in Shallow Water

a. Standard Shallow Water Model

b. Adapted Shallow Water Model

i

. One-Dimensional Waves

ii. Two-Dimensional Waves

3.

Airy’s Linear Wave Theorya. Standard Airy’s

Linear Wave Theoryb. Generalised Airy’s Linear Wave Theory4. Discussions5. ConclusionsSlide49

Discussion:Small scattered wiggles

The small wiggles seen in positive gravity perturbation

Instantaneous difference, 1D

Gravity perturbation vs no perturbation

Instantaneous difference, 2D

Gravity perturbation vs no perturbation

Small wiggles

Possibly explained by the diagnostic formalism (Schrodinger question)?

Energy level; Wave trappings?Slide50

Discussion: Experimental Validation of Adapted Shallow Water Model

Replacing Gravity by Electromagnetic force?

Direct

Navier

Stokes Simulation?Slide51

Discussion:Airy’s Linear Waves in 3D Space

Three-Dimensional Gravity Field

Conformal coordinates (q

1

, q

2

, q

3)?

Analytical solution?Numerical simulation of governing equations?Slide52

Content Page

1. Background - Gravitational Fields and Surface Fluid Waves

2. Waves in Shallow Water

a. Standard Shallow Water Model

b. Adapted Shallow Water Model

i

. One-Dimensional Waves

ii. Two-Dimensional Waves

3.

Airy’s Linear Wave Theorya. Standard Airy’s

Linear Wave Theoryb. Generalised Airy’s Linear Wave Theory4. Further Discussions5. ConclusionsSlide53

Conclusions1. Adapted Shallow Water Model

Gravity -> Amplitude, Wavenumber and Scattering

In practice: effects on ocean surface waves by

Gravity

: Unobservable

MSL

: Observable, sufficient by standard model

2. Generalised

Airy’s Linear Wave Theory

Mapping of conservative gravity field into uniform field Redefine ‘depth’ and ‘horizontal’ coordinateOnly valid for one-dimensional wavesConsistent with Adapted Shallow Water ModelSlide54

Things omitted…Effective gravity field due to rotation

Detailed scale analysis in adapted shallow water model

Potential vorticity in adapted shallow water model

Length scales of perturbation on wave transmission and scattering

Limitations of 2D gravitational potential

Attempts on 3D generalised

Airy’s

linear wave theory

and many more…Slide55

Thank you!

Questions are welcome!Slide56

END