Surface waves –wind ,
groups, hurricanes and
tsunamis.
INI Waves Symposium Thursday july
17
Julian Hunt
UCL. ASU, TUD, Cambridge
Thanks Sajjadi, Klettner
Grimshaw,Chow
Sajjadi S G, Hunt JCR , Drullion F (2013)
Asymptotic Multi-layer analysis of wind over unsteady
monochromatic surface waves
J.Eng Maths
Note critical layer at zc where U(z)= cr , wave speed. In above
profile zc is in shear stress layer (zc < l s) . Also shown is
where (zc > l s )for larger c r-as analysed for c i >0 by
Miles(1957)
Waves and Interfaces
Seminar at UCL
Julian Hunt
1.Wind-Wave Coupling
Introduction
Physical mechanisms involved for wind
over unsteady surface and wave groups.
Triple(or quadruple) deck theory for
turbulent shear flows.
Combining sheltering and unsteady critical
layer mechanisms.
Explain why groups are most efficient
mechanism for air-sea energy exchange.
Through analytical and computational
modeling.
Schematic
Separated Sheltering L = O(a).
Non-separated Sheltering.
Schematic of wind over wave mechanisms for steady low amplitude waves.
Shows sheltering mechanism in the surface layer and its coupling with the outer
flow.
Steady waves (ci=0). Benjamin, Townsend, BH,
Mastenbroek… non-separated sheltering . But critical
layer is important ; drag is affected for c r >u*
Unsteady Waves
For unsteady waves where 0 < c i /U* ~ 1.
Mean streamlines of flow over the waves are
shown in a frame moving with the waves.
Waves closed loops are centred at the elevated
critical height.
Profiles of u–velocity perturbation for inviscid
flow (when ν e = 0) is shown below.
Perturbations become singular as the growth
rate c I -> 0 .
In phase
Vel
perturbat
ion
Waves growing; exp (i(k (x-c rt) .exp(-k ci t) . Inviscid flow
Note singular behaviour as ci ->0, near critical layer
Out of
phase
Vel
perturb
ation
Inviscid Mechanism for Unsteady Waves
Fig of c I vs kz shows as ci  0 out-of-phase
perturbation to \Delta u becomes very large.
Occurs in very thin layer of thickness of O(c_i).
Vorticity ( ωy) amplifies on lee-side and
reduced on upwind-side.
Leads to mean stream lines being deflected,
"lower pressure" on the lee side  "higher
drag".
when ci > 0, wave grows; ci <0 wave decays.
But there is a net force on the wave produced
by the integrated effect of the critical layer as
|ci|->0. ( Miles did not calculate profiles-only
the integral effect prop to U’’(at z=zc))
Historical Conclusions
Miles (1957)/Lighthill (1962) calculate energy
input/ growth for ci -> 0; a = const:; ka << 1.
Note Lighthill’s analysis with delta functions
(p192) suggests importance of thin , singular
critical layers –but they were not mentioned
explicitly.
They conclude: there is a net inviscid force on
monochromatic non-growing waves (steady
waves).
Computations (Steady Waves)
We adopt full realizable RSM (TCL) [Sajjadi,
Craft & Feng].
We compare full Reynolds-stress model with
DNS, Sullivan et al.
Also adopted semi-implicit FD solver with LRR
turbulence model for comparison.
Waves closed loops are centred at the critical
height.
Very similar behaviour as DNS.
Computational Domain
Schematic computational domain (84 x 50 x 5)
mesh points.
Bulk Reynolds number Re = 8000.
Fully implicit, collocated, general curvilinear
coordinates finite volume.
Coupled to water through orbital velocities.
Energy Transfer from Wind to Waves (I)
Total energy transfer parameter, \beta.
Critical layer and sheltering mechanisms for unsteady
waves (ci ) ( << U*) vs wave age cr/U1.
+++++, Miles (1957) calculation (ci = 0; νe = 0)
Thick solid line, Janssen (1991) parameterization of Miles
(1957) for ci = 0 ; inviscid ; zero eddy viscosity
Thin solid line, present formulation: (\beta_T + \beta_c) for
ci >0 ; finite eddy viscosity .
\circ, Numerical simulation using Reynolds-stress closure
model for ci >0 ; finite Reynolds stress.
Monochromatic waves .Growth rate against wave speed / wind
speed –models , and data
Note how β rises with moderate increase in crit layer thickening of
Inner viscous layer. But when cr/Ui ~5, the inner viscous layer diminishes the
Sheltering mechanism , and β drops.
Monochromatic waves .Wave growth based on Miles model ,
with zero and finite eddy viscosity. Note no growth in latter case
Wave Groups (first proposed (?)by
M.E.McIntyre –DAMTP )
On going project.
Above application to group of waves
Individual waves grow within group on
upwind side and decrease on downwind
side (hence plus/minus inertial critical layer
effects). But shear layer over wave group
leads to zc higher on lee side , U’’(zc)
smaller. Hypothesis is net positive effect
Study effect of non-uniform critical layer
over groups (analytically &
computationally).
Wave group -3 -5 waves; concept of double effect of sheltering +
Inertial critical layer ; numerical simulations
Conclusions
Physical growth mechanism is NOT soley
due to steady Critical layer CL.
Critical layer is unsteady problem.
Growing/decaying wave amplitude in the
groups increases CL on the downside of
group where wave shape changes.
CL plays an important role on sheltering.
Asymmetrical sheltering leads to reduction
in wind speed.
2. Tsunami waves
The problem of negative tsunami waves travelling over
long distances , being transformed into positive waves
and impacting on beaches , before or after the transition.
Ref .Klettner, C.A., Balasubramanian, S., Hunt, J.C.R.,
Fernando, H.J.S., Voropayev, S.I. and Eames, I., 2012
Draw down and run-up of tsunami waves on
sloping beaches. Proc. Inst. Civil Engineers-Engineering
and Computational Mechanics 165,119-129 .
Earthqua
ke off jap
March
2011
Note
depressio
n to north
Japan
Tsunami
(Neg wave to
North of Sendai)
Note maximum
near Sendai
Opposite
Centre of Earth
Quake.
Note blue
Inundation
inland..
Measured data around the Pacific coast of Japan. Red and blue
color bars indicate inundation height and run-up height. (a) Japan
view, (b) Sendai Plain.
Negative Tsunami wave arriving at
japan coast 2011
•
quaasi steady shallow water analysis
• c = sqrt (gh (x))(1 +/- ½ a(x.t) /h(x))
• h(x)= ho –zs. (local water depth)
• Δc = ½ ([-/+ α w(x.t) – α b] sqrt(g/ho)
• α b = dzs /dx. Let dzs/dt =+/- α w(x.t)
Sloping water surface ;
• a(x,t)=+/-| α w(x.t) |( x f – x)
• -d| α w(x.t) |/dt =
•
½ ( +/- | α w(x.t) | - α b ) sqrt(g/ho)
SRI LANKA 2004—very large negative tsunami waves on beaches
Caused very sudden and large loadings on buildings-
Backward Tsunami slope
•
•
•
•
•
on the slope
Where x< xmx , alpaw <0
Leading slope where alphaw <0
Surface slope ~ bottom slope .
On trailing V shaped slop Wave slope
|alphaw| tends to increase
• -> rapid change in rear wave shape
• -> shortening of wave dL/dt <0->
• Larger amplitude wave , and force
momentum at beach
Negative tsunami on a
beach,tested in ASU hyd. lab
6.92m
7.92m
8.92m
9.92m
11.9m
12.9m
14.9m
15.9m
6
2
(a)
0
a
(-)
(cm)
4
-2
-4
-6
0
50
100
150
xF - x (cm)
10
6.92m
8
7.92m
8.92m
6
(cm)
(-)
a
(b)
9.92m
11.9m
4
13.9m
2
14.9m
0
-2
-4
-6
-8
0
50
100
150
xF - x (cm)
15
6.92m
7.92m
8.92m
9.92m
10.9m
11.9m
12.9m
13.9m
10
a(-) (cm)
• Note the sharp rise on
the downstream end
behind the deepening
depression-same n-l
process as for + waves.
• Depression leads to
shore line retreat
• large wave up the
beach .
5
(c)
0
-5
-10
0
50
100
xF - x (cm)
150
200
Asu Lab expt for negative tsunami
How to explqain this in a single picture/notice ?
Up=down solitary wave moving to
right t1 -modelled by non-linear
mod.KdV eqn
KWChow, R.Grimshaw
Ut+6UUUx+Uxxx=0 -> exact soln
(~num results)
------>
Up-down solitary wave moving
to right (t2)
Note amplitude rises faster –but length scale
Reduces
Up-down solitary wave at
critical time tc-large amplitude
Hypothesis of S Day-may be cause
Of exceptional tsunami events on mid
ocean islands?
X ->
up-down wave to right -after critical moment –lower
Amplitude t4>tc
Up-down wave long after critical time. Note equal amplitude up=down.
-note moving to right
Note warming and less ice in Arctic ocean affects
Wind driven Waves , which break up ice and leads to
Further warming (Wadhams) , and
Less ice , plus seismic activity , is likely to
Lead to tsunamis on very flat arctic coastline
Concetn to local communities and to extractive industries
And to possible transport dangers
-
Wind structure 0f tropical cyclones using new data from met towers on south
Up-down wave long after critical time. Note equal amplitude up=down.
Liu, Kareem, Hunt
-note moving to right
(a)
(a)
Mean wind speed profiles at Delat Island tower in tropical cyclone Nuri
Mean wind speed profiles at Zhizai tower in tropical cyclone Hagupit
Fig. 6 Mean wind speed profiles at the different radial positions (defined in Fig. 5) for the
tropical cyclones Nuri and Hagupit. Note the pronounced low level wind speed maxima
for the positions near the eyewall regions (FEW, BEW), and less pronounced in the outer
regions (FOV, BOV)
Mean wind profiles
(a)
(a)
Mean wind speed profiles at Delat Island tower in tropical cyclone Nuri
Mean wind speed profiles at Zhizai tower in tropical cyclone Hagupit
Fig. 6 Mean wind speed profiles at the different radial positions (defined in Fig. 5) for the
tropical cyclones Nuri and Hagupit. Note the pronounced low level wind speed maxima
for the positions near the eyewall regions (FEW, BEW), and less pronounced in the outer
regions (FOV, BOV)
Spectra at outer radius
Near eye of vortex