Numerical simulation of internal waves in the coastal ocean

1
Numerical simulation of internal waves in the
coastal ocean

• Oliver Fringer
• Environmental Fluid Mechanics Laboratory
• Department of Civil and Environmental Engineering
  
• Stanford University
• 11 April 2006

2
Tidal dissipation in the worlds oceans
What percentage of tidal energy that is lost at
ocean boundaries is converted into internal
tidal energy?
TIDES
Open ocean mixing dissipation 75
Topographic features 25
1 TW
Mixing dissipation X?
Internal tides X?
Mixing dissipation
3
Internal Waves O(104 m)
Straight of Gibraltar
Baja California
Image http//cimss.ssec.wisc.edu/goes/misc/010930
/010930_vis_anim.html
Image http//envisat.esa.int/instruments/images/g
ibraltar_int_wave.gif
4
Internal Waves O(101 m)
Klymak Moum, 2003
Venayagamoorthy Fringer, 2004
5
Internal waves O(10-1 m)

• length 3.0 m, width 0.2 m, depth 0.5 m

6
Motivation

• Internal waves are believed to account for a
  significant portion of mixing and dissipation
  within the oceans, especially near complex
  bathymetry
• Where are they being generated and where are they
  breaking?
• How do they affect pollutant transport?
• How do they affect the propagation of acoustic
  signals?

7
SUNTANS Overview

• SUNTANS
• Stanford
• Unstructured
• Nonhydrostatic
• Terrain-following
• Adaptive
• Navier-Stokes
• Simulator
• Finite-volume prisms
• Parallel computing C MPI ParMetis
• For details see Fringer et al. (2006)Ocean
  Modelling, in press.

Grid B. Wang
8
Grid structure
Side view

• Unstructured in plan.
• Structured, or z-levels, invertical.

Top view
z-levels
Serial
Parallel
9
Parallel Graph Partitioning

• Given a graph, partition with the following
  constraints
• Balance the workload
• All processors should perform the same amount of
  work.
• Each graph node is weighted by the local depth.
• Minimize the number of edge cuts
• Processors must communicate information at
  interprocessor boundaries.
• Graph partitioning must minimize the number of
  edge cuts in order to minimize cost of
  communication.

Delaunay edges Voronoi graph
Voronoi graph of Monterey Bay
10
ParMetis Parallel Unstructured Graph
Partitioning (Karypis et al., U. Minnesota)
Five-processor partitioning Workloads 20.0
20.2 19.4 20.2 20.2
Original 1089-node graph of Monterey Bay, CA
Use the depths as weights for the workload
11
Bandwidth reduction via graph ordering
Consider the simple triangulation shown
12
ParMetis Parallel Unstructured Graph Ordering
(Karypis et al., U. Minnesota)
Unordered Monterey graph with 1089 nodes
Ordered Monterey graph with 1089 nodes
Ordering increases per-processor performance by
up to 20
13
Pressure-Split Algorithm

• Pressure is split into its hydrostatic and
  hydrodynamic components
• Hydrostatic pressure

Surface pressure
Barotropic pressure
Baroclinic pressure
14
Boussinesq Navier-Stokes Equations with
pressure-splitting
Surface pressure gradient
Internal waves c O ( 1 m/s )
Surface waves c O ( 100 m/s )
Acceleration uO(0.1 m/s)
15
Semi-implicit time-advancement scheme

• First step hydrostatic pressure 2D Poisson
  equation for h
• Second step nonhydrostatic correction 3D
  Poisson equation for
• Is it necessary to compute the nonhydrostatic
  pressure?

16
Hydrostatic vs. Nonhydrostatic flows

• Most environmental flows are Hydrostatic
• Hyperbolic character
• Long horizontal length scales, i.e. long waves
• Only in small regions is the flow Nonhydrostatic
• Elliptic character
• Short length scales relative to depth

Long wave (hydrostatic)
free surface
Steep bathymetry (nonhydrostatic)
bottom
17
When is a flow nonhydrostatic?
Aspect Ratio
18
(No Transcript)
19
When are internal waves nonhydrostatic?
CPU time 1 day CPU time 3 days
20
Hydrostatic vs. Nonhydrostatic lock exchange
computation
Hydrostatic
Nonhydrostatic
Doman size 0.8 m by 0.1 m (grid 400 by 100)
21
Conditioning of the Pressure-Poisson equation

• The 2D x-z Poisson equation is given by
• For large aspect ratio flows,
• To a good approximation,
• The preconditioned equation is thenwhich is a
  block-diagonal preconditioner.

22
Speedup with the preconditionerwhen applied to a
domain with dD/L0.01
No preconditioner (22.8X)
Diagonal (8.5X)
Block-diagonal (1 X)
23
Other features required when wetting and drying
is employed
Wetting and drying incurs small cell heights
Implicit vertical advection/diffusion
24
Monterey Bay An internal wave sanctuary
25
Internal tide generation over topography
Deep ocean (3000 m deep)
Shelf (500 m)
Depth
Density
26
Internal wave generation in Monterey Bay

• Horizontal resolution 300 m
• Boundary conditions
• OSU Tidal Inversion Software M2 tidal velocites
  at boundaries
• Initial density field
• Average of 50 CTD casts (Petruncio, et al., 2002)
• Bottom drag Cd0.005
• Constant vertical eddy viscosity of 2 X 10-3 m2
  s-1
• Constant horizontal eddy viscosity of 20 m2 s-1
• No scalar diffusivity
• 16 Processors, 4 million grid cells, 4.8
  seconds/time step (2X real time)

27
Three-dimensional internal wave generation
28
The effect of grid resolution on internal wave
generation
It's all in the GRID!!!
Grid size 3000 m (80K cells) ? Umax 2
cm/s Grid size 300 m (4M cells) ? Umax 16.1
cm/s Grid size 60 m (45M cells) ? Umax ?
Field data from Petruncio et al. (1998) ITEX1
Mooring A2
SUNTANS results w/300 m grid Max U 16.1 cm/s
Simulation time 8 M2 Tides
U velocity (cm/s)
Figures Steven Jachec
29
Where are internal waves generated?
Energy flux
Across 1000 m of water,
E 1 MW
1 KW/m
Figure S. Jachec, EFML
30
log10(W/m2)
52 MW
Figures S. Jachec, EFML
31
Global tidal energy budget

• World coastline 532,000 km
• Monterey Bay coastline 100 km
• Internal tidal generation in Monterey Bay 52 MW
• Scaled to global coastline 0.27 TW

32
Acknowledgments

• Collaborators
• Prof. Margot Gerritsen, Prof. Bob Street
• Field data E. Petruncio, L. Rosenfeld, J. Paduan
• Students S. Jachec, K. Venayagamoorthy, B. Wang
• Support
• ONR Grants N00014-02-1-0204, N00014-05-1-0294,
  NSF Grant 0113111
• For more information visit http//suntans.stanfo
  rd.edu