1. Modelling Overwash of Ice Floes
Presented by:
David Skene – PhD Student, University of Adelaide
Supervised by:
Dr Luke Bennetts – University of Adelaide
Accoc. Prof. Mike Meylan – University of Newcastle

2. Motivation: Background
Sea Ice Coverage
Sea ice coverage is in a constant process of decay and growth
Due to multiple environmental effects
Antarctica Feb 2013
Antarctica Sept 2013
Antarctica Feb 2014
Antarctica Sept 2014
Figures taken from NSIDC (2015)

3. The Marginal Ice Zone (MIZ)
Motivation: Background
The Marginal Ice Zone (MIZ)
Interface between open ocean and frozen ocean
“the part of the ice cover which is close enough to the open ocean boundary to be affected by its presence”
Width of kms
Contains thin O(cm) but long O(10-100m) ice sheets called floes
The Marginal Ice Zone
Open Ocean
Pack Ice

4. Observations and Measurements in the MIZ
Motivation: Background
Observations and Measurements in the MIZ
Pioneering observations made by Wadhams et al in 1970s (eg. Wadhams et al 1975)
Experimental work in the MIZ is difficult due to expense, safety, and expertise required
Lack of experimental work until Kohout et al’s 2014 publication in Nature
Investigated wave-ice interactions in MIZ
The Marginal Ice Zone

5. Wave’s Effects on Sea Ice (Kohout et al 2014)
Motivation: Background
Wave’s Effects on Sea Ice (Kohout et al 2014)
Measured trends in ice coverage by satellite
Compared to model of significant wave amplitude trends
Found correlation between ice coverage and wave activity
Blue: Trend in wave activity
Orange: Trend in ice coverage
Figure from
Kohout et al 2014

6. Wave Attenuation and Floes (Meylan et al 2014)
Motivation: Background
Wave Attenuation: Loss in wave intensity through a medium
Greater attenuation coefficient => Decay more over distance
Shorter waves propagate less distance through the MIZ
Kohout’s measurements found the MIZ acts as a ‘Low-Pass Filter’
Consistent with current models
Plot of Wavelength vs Attenuation
Low-pass filter
Attenuation Coefficient
Figure from
Meylan et al 2014
Wavelength

7. Wave Attenuation and Amplitude (Meylan et al 2014)
Motivation: Background
Measurements found attenuation also dependant on wave amplitude
Particularly noticeable for small wavelengths
(Still could be present for large wavelengths)
Current modelling based on wave scattering
Scattering suggests attenuation is independent of amplitude
Plot of Amplitude vs Attenuation
Small wavelength
Attenuation Coefficient
Larger wavelength
Figure from
Meylan et al 2014
Wave Amplitude

8. More Data: Wave Tank Experiments
Motivation: Wave Tank Experiments
More Data: Wave Tank Experiments
Recent years: Experiments in wave tanks to validate ice floe models (Montiel et al 2012; Bennetts & Williams 2014; Meylan et al 2015)
Used thin plastic plates as model floes
Frequent observation has been what we call overwash
No Overwash
Overwash
Figures from
Bennetts & Williams 2014

9. Motivation: Wave Tank Experiments
What is Overwash?
Plate oscillations causes edges to dip into water
This dipping causes fluid to wash over the top
We call this process overwash
It is not included in theoretical models
Overwash Fluid
Plate’s Edge Dips into Water

10. Theoretical Modelling of Waves and Floes
Motivation: Wave Tank Experiments
Theoretical Modelling of Waves and Floes
Developed since the 1970s
Wave models built on solitary floes (Squire et al 1995, 2007)
I’ll refer to as: Linear Potential Theory
Plate is modelled as a thin, long, and elastic
Determines the scattering effect floes produce
Theory is linear with respect to amplitude
Conserves energy
Floe

11. Overwash and Wave Transmission
Motivation: Wave Tank Experiments
Overwash and Wave Transmission
Experiments conducted at the University of Melbourne
Compared:
Theoretical transmission due to linear potential theory (Dashed Line)
With experimental transmission (Box and Wisker)
Show deviation for greater steepness
Plots of Transmitted vs Incident Steepness
Short Wavelength
Long Wavelength
Transmitted Steepness
Incident Steepness
Incident Steepness
Figures from
Luke Bennetts

12. Focus of my PhD Motivation: My Work
It appears overwash has some effect on wave propagation. It also appears it’s highly non-linear.
Goals:
Develop a theoretical model for overwash
Determine when and how overwash occurs
Determine overwash’s effect on wave transmission.
Given overwash’s non-linearity Can this explain transmission’s dependence on incident wave amplitude?

13. Experiment to Observe and Quantify Overwash
Overwash Experiments
Experiment to Observe and Quantify Overwash
Wave tank testing conducted at Plymouth University, UK
1m square floating PVC and Polypropylene was used as floes
Thicknesses of 5mm – 40mm
Regular incident waves of varying Wavelength and Steepness
Overwash recorded via depth probe and camera

14. Example Overwash Video
Overwash Experiments
Example Overwash Video
Overwash Bores
Incident
Wave Direction

15. A Key Result from these Experiments
Overwash Experiments
A Key Result from these Experiments
Parallel experiments measuring plate motion
Compared motion predicted by linear potential theory to experimental motion
Showed the plate moves according to linear potential theory
Motion of Plate Comparison
Figures from
Luke Bennetts

16. 2D Model of Overwash Key Modelling Assumptions: Theoretical Model
Linear Potential Theory models plate and surrounding water
Shallow Water Equations models overwash
Overwash has a negligible effect on the surrounding domain
Linear Potential Theory drives the Shallow Water Equations using one-way coupling
Shallow Water Equations
One Way Coupling
Incident Wave Direction
Linear Potential Theory

17. Linear Potential Theory (Incident Wave from Left to Right)
Theoretical Model: Linear Potential Theory
Assumptions:
Water as inviscid incompressible irrotational fluid
Plate modelled as Euler-Bernoulli beam
Plate has no lateral drift
Linearization of surfaces
No overwash effects
This Creates a Boundary Value Problem
Determine:
Plate motion
Surrounding wave field
Solved in the frequency domain i.e.
Reflected Wave
Transmitted Wave
Incident Wave Direction
Linear Potential Theory

18. Theoretical Model: Linear Potential Theory
Mathematical Conditions
Fluid is modelled via velocity potential φ where
In the Frequency domain φ must satisfy the following conditions:
For some frequency ω
Radiation conditions & fluid must match plate motion at contact
Laplace’s Equation:
No Bed Penetration:
Linearised Free Surface:

19. Step 1: Motion of the plate
Theoretical Model: Linear Potential Theory
Step 1: Motion of the plate
Motion determine via Kirchoff-Love plate theory
Split into motion into orthogonal modes wn
Rigid body modes:
Modes of vibration
For Eigenvalues:
Weights ξn (constants) need to be determined via coupling

20. Theoretical Model: Linear Potential Theory
Step 2: Splitting the Potential into Components
To help solve we make
φi is the incident wave that forces the system:
With amplitude A, frequency ω, and k0 real
φD is the potential under the plate where the plate is stationary:
φR is the potential under the plate where the plate moves by the nth mode:
Otherwise, all of these satisfy the previous conditions

21. Theoretical Model: Linear Potential Theory
Step 3: Use of a Green’s Function
To solve for the diffracted and radiation potential define:
This corresponds to the fluid being forced at some point x0
It is analytic but complicated
Integrating around the entire fluid domain (via Gauss’ Divergence Theorem) gives:
On z=0
Means the potential can be solved at points via numerical integration

22. Theoretical Model: Linear Potential Theory
Final Step: Determine the modal weights
Use the previously unmentioned dynamics condition under the plate to give:
β is the non-dimensional stiffness, ϒ is the non-dimensional mass
Multiply by wm, take the inner product, truncate to N modes of vibration to give a matrix system
System determines the modal weights ξn
Can also use the Green’s function method to determine radiated and transmitted potentials around the plate

23. Modelling the Overwash Domain
Theoretical Model: Shallow Water Equations
Modelling the Overwash Domain
Observations of Overwash Showed:
Thin relative to length
Bores
Consistent with:
Shallow Water Equations
Except
The shallow water equations require stationary bed
Overash has a slowly accelerating bed

24. Slowly Accelerating Bed
Theoretical Model: Shallow Water Equations
Slowly Accelerating Bed
Shallow water equations assume no penetration condition on the bed (on z=zb)
But when the bed moves:
Same steps for deriving the shallow water equations and the assumption:
Results in:
The shallow water equations

25. The Shallow Water Equations
Theoretical Model: Shallow Water Equations
The Shallow Water Equations
The shallow water equations are a set of non-linear hyperbolic PDEs
They cannot be solved readily in the frequency domain
Require a numerical time-stepping solution with appropriate boundary conditions
Even time-stepping requires care in dealing with
Bores that appear as shocks
Creation of unnatural extrema
Figure from Wikipedia

26. A Time-Stepping Method
Theoretical Model: Shallow Water Equations
A Time-Stepping Method
A suitable time-stepping scheme developed from Kurganov & Tamdor (2000) and Kurganov et al (2001)
I shall spare you the details but:
Step 1: Discretise the x domain into M points
Step 2: Discretise the x-derivative at each jth point:
Step 3: Apply appropriate total variation diminishing (TVD) time stepping method to discretise time derivative into the future at t=t2>t1
I used TVD RK2

27. Theoretical Model: Coupling
Coupling of Domains
Ideally, I’d like the two domains to be fully coupled.
Difference between frequency domain and time stepping makes this difficult
Instead, for this model:
The heights and velocity at the plate’s edges are determined by Linear Potential Theory
These provide the boundary conditions for the Shallow Water Equations

28. Theoretical Model: Summary
Linear Potential Theory determine motion of plate and surrounding fluid for all time
Shallow water equations determine motion of overwash fluid at some future time given present time with BCs from Linear Potential Theory
Model is complete

29. Comparison of Theoretical Model and Experiments
Both experiments and model driven until a quasi-steady state
Experiments found no overwash at centre in 19/83 cases
Theoretical model found no overwash 17/83 cases
In the two different cases, overwash did occur just did not wet the whole plate
Suggests linear potential theory predicts occurrence of overwash well

30. Video of Comparison (Moderate Overwash) 1
Model and Experiments

31. Video of Comparison (Moderate Overwash) 2
Model and Experiments

32. Comparison of Moderate Overwash Depth Variation
Model and Experiments
Comparison of Moderate Overwash Depth Variation
Video 1
Video 2
Centre Depth Variation (mm)
Time (s)
Time (s)
Experiment in Red, Theoretical Model in Blue

33. Video of Comparison (Small Overwash)
Model and Experiments

34. Comparison of Small Overwash Depth Variation
Model and Experiments
Comparison of Small Overwash Depth Variation
Centre Depth Variation (mm)
Time (s)
Experiment in Red, Theoretical Model in Blue

35. Comparison of Overwash Average and Deviation
Model and Experiments
Comparison of Overwash Average and Deviation
Results for 10mm Thick PVC
x-axis varies with incident wave properties
Experiment in Red, Theoretical Model in Blue

36. Comparison of Overwash Average and Deviation
Model and Experiments
Comparison of Overwash Average and Deviation
Results for 19mm Thick PVC
x-axis varies with incident wave properties
Experiment in Red, Theoretical Model in Blue

37. Video of Comparison For High Steepness and Wavelength
Model and Experiments
Video of Comparison For High Steepness and Wavelength

38. Causes of Disagreement
Model and Experiments
Causes of Disagreement
Coupling effect at plate’s edge
Breaking of waves
Difficult to include
Centre Depth Variation (mm)
Time (s)
Experiment in Red, Theoretical Model in Blue

39. Results Summary Summary
Linear Potential Theory predicts when overwash will occur
One-way coupling predicts overwash well for short wavelength low steepness waves where small and moderate amounts of overwash occur
Additional effects cause discrepancy for long wavelength high steepness

40. Future work Summary Incorporate coupling into the model
Extend shallow water domain further for wave breaking
Extend into 3D
Analyse effect on wave transmission

41. Acknowledgements Dr Luke Bennetts – University of Adelaide
Accoc. Prof. Mike Meylan – University of Newcastle
Assoc. Prof. Alessandro Toffoli – Swinburne University of Technology
The University of Adelaide
Australian Antarctic Division

42. References
L. Bennetts & T.Williams, 2015, Water wave transmission by an array of floating disks, Proceedings of the Royal Society of London, Series A; 471
S. Godunov, 1954, Ph.D. Dissertation: Different Methods for Shock Waves, Moscow State University
A. Kohout, M. Williams, S. Dean, M. Meylan, 2014, Storm-induced sea-ice breakup and the implications for ice extent, Nature, 509;
A. Kurganov & E. Tamdor, 2000, New high-resolution schemes for nonlinear conservation laws and convection-diffusion equations, Journal of Computational Physics, 160(1);
A. Kurganov, S. Noelle, G. Petrova, 2001, Semidiscrete central-upwind schemes for hyperbolic conservations laws and hamilton-jacobi equations, SIAM Journal on Scientific Computing 23(3);
M. Meylan, L. Bennetts, A. Kohout, 2014, In-situ measurements and analysis of ocean waves in the Antarctic marginal ice zone, Geophysical Research Letters, 41;
M. Meylan, L. Bennetts, A. Alberello, C. Cavaliere, A. Toffloi, 2015, Experimental and theoretical models of wave-induced flexure of a sea ice floe, Physics of fluids Letters, accepted
F. Montiel, L. Bennetts, V. Squire, 2012, The transient response of floating elastic plates to wavemaker forcing in two dimensions, Journal of Fluids and Structures, 28;
V. Squire, J. Dugan, P. Wadhams, 1995, Of ocean waves and sea ice, Annual Review of Fluid Mechanics, 27(1);
V. Squire, 2007, Of ocean waves and sea-ice revisited, Cold Regions Science and Technology, 49(2);
P. Wadhams, 1975, Airborne laser profiling of swell in an open ice field, Journal of Geophysical Research, 80(33),