1
Forecasting wave conditions in
cyclones using adaptive methods
Richard Gorman, Stéphane Popinet &
Doug Ramsay
National Institute of Water & Atmospheric
Research
• Introduction to wave modelling
• Gerris, an adaptive flow solver
• Spectral wave models
• Initial developments and applications
• Future work
Contents
2
• Global wave models account for
waves generated locally, and by
distant storms, propagating
across oceans, interacting with
topography, dissipating
• Forecasting → 6 days
• Archive:
• Wave climate
• Wave extremes
Introduction
Analysis Forecast
Present time
20-yr wave hindcast
wave height correlation with SOI
Negative: higher with El Niño; Positive: higher with La Niña
3
• Need to work across multiple
scales, from global to local
• Presently use nested cartesian
grids, with some developments
on irregular meshes
• Global scale, e.g.: 1.25º long, 1º
lat (NOAA/NCEP Wavewatch
grid).
Scales
Wind field
resolution:
12 km
Wave model
resolution:
12 km
Vector spacing:
12 km
Spatial scale effects
Wind field
resolution:
12 km
Wave model
resolution:
1 km
Vector spacing:
4 km
4
•
Cyclone Waka:
•
Peak V
max
= 50 m/s, R
max
= 40
km, V
storm
= 5-10 m/s
•
Weather system radius
~ 2000 km
Spatial scale of wind fields
•
Kosrae: 8 December
2008
•
Tokelau: Tropical cyclone
Percy, 25 February 2005
5
Gerris adaptive flow solver
•
Solves the time-
dependent Navier-
Stokes equations
•
Solves for flow, and the
advection of tracers
•
Quadtree grid in 2D
(octree in 3D)
•
Adaptive mesh
refinement: the
resolution is adapted
dynamically to the
features of the flow
Spectral wave modelling
6
• e.g. WAM, Wavewatch, SWAN
• describe the evolution of the wave action density
• σ = wave frequency,
θ = wave direction
• x, y = spatial coordinates, t = time
• Transport equation:
Spectral wave models
)
,
,
,
,
(
t
y
x
N
θ
σ
(
)
(
)
(
)
(
)
σ
θ
∂
∂
σ
∂
∂
∂
∂
∂
∂
∂
∂
θ
σ
S
N
C
N
C
N
C
y
N
C
x
t
N
y
x
=
+
+
+
+
Advection = wave energy moving at the group velocity
(C
x
,C
y
)
Refraction = changing frequency and direction
of waves due to currents, topography
S
= source terms for wind input, dissipation, nonlinear interactions
Wave modelling in Gerris
• Gerris already solves for velocity, and advects velocity and
other quantities on an adaptive quadtree grid
• Modify to advect wave spectral components, with known
velocities
• At present, no treatment of refraction
• Source term processes (refraction, wind input, dissipation,
nonlinear interactions) computed by calls to Wavewatch III
(
)
(
)
(
)
(
)
σ
θ
∂
∂
σ
∂
∂
∂
∂
∂
∂
∂
∂
θ
σ
S
N
C
N
C
N
C
y
N
C
x
t
N
y
x
=
+
+
+
+
7
Simulating an artificial “cyclone”
•
The centre of the cyclone is moving
south at a speed of 555 km/day.
•
Maximum wind speed V
max
is at a
radius R
max
= 100 km from the centre
•
V
max
= 0 initially, increasing to 50 m/s
after 25 hours, then constant
•
The wind speed at a distance r from
the centre is given by:
⎟
⎠
⎞
⎜
⎝
⎛ −
⎟
⎠
⎞
⎜
⎝
⎛
=
r
R
e
r
R
t
V
t
r
V
max
1
2
2
max
max
)
(
)
,
(
Resulting significant wave height
12 hours
24 hours
36 hours
48 hours
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Adaptive mesh (48 hours)
6.5 km
13 km
26 km
52 km
104 km
208 km
Spatial resolution
Model intercomparison
• Maximum significant wave height in the domain, as a function of time
• Standard Wavewatch III model (fixed resolution)
• Gerris/Wavewatch model in adaptive and fixed resolution
• “GSE alleviation” = smoothing between direction bins
9
Run times (seconds)
Two paths to an adaptive-grid wave model
•
Start with Gerris, which
already solves for velocity,
and can advect velocity and
other quantities on an
adaptive quadtree grid
•
Modify Gerris to advect wave
spectral components, with
known velocities
•
The resulting model has
been successfully applied to
a standard test case
•
Shows the benefits of
trading off spatial and
spectral resolution
•
Start with
Wavewatch, which
already solves for all
wave processes on a
cartesian grid
•
Modify Wavewatch
to handle spatial
advection on a static
quadtree grid
•
Other processes are
local, hence
unaffected
10
Summary
• The existing Gerris flow solver has been modified to
handle spatial propagation of spectral wave energy on a
fully adaptive grid
• It is coupled to standard by calling Wavewatch III code to
handle other “local” processes (wind input, dissipation,
nonlinear interactions)
• It has been applied to a test case simulating wave
generation under an artificial “cyclone”, showing the run-
time benefits of variable spatial resolution
• The results are consistent with those of standard
Wavewatch III
• The use of variable spatial resolution results in efficiency
gains of over an order of magnitude
Immediate future…
• Further tests are planned for more realistic cyclone
simulations and involving interaction with shallow water
areas (hopefully Tonga)
• Application has implications for:
– More accurate wave forecasting of cyclone wave conditions
within Pacific-wide wave models
– Improved accuracy of probabilistic cyclone wave conditions for
any particular location through ability to efficiently run many
more cyclone characteristic scenarios
– Improving representation of cyclone wave translation over reef
flats and shallow areas (assuming availably bathy!)
• Work also planned to look at storm surge / wave
coupling over reef flat systems