Numerical methods for coastal ocean modeling

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Numerical methods for coastal ocean modeling

• Oliver Fringer
• Environmental Fluid Mechanics Laboratory
• Department of Civil and Environmental Engineering
  
• Stanford University
• 3 March 2006

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Tidal dissipation in the worlds oceans
Roughly 25-30 of tidal energy is dissipated near
topographic features (about 1 TW!).
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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
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Internal Waves O(101 m)
Klymak Moum, 2003
Venayagamoorthy Fringer, 2004
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Internal waves O(10-1 m)

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

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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?

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SUNTANS Overview

• SUNTANS
• Stanford
• Unstructured
• Nonhydrostatic
• Terrain-following
• Adaptive
• Navier-Stokes
• Simulator
• Finite-volume prisms
• Parallel computing MPI ParMetis

Side view
Top view
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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
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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
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Bandwidth reduction via graph ordering
Consider the simple triangulation shown
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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
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Pressure-Split Algorithm

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

Surface pressure
Barotropic pressure
Baroclinic pressure
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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)
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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?

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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
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When is a flow nonhydrostatic?
Aspect Ratio
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(No Transcript)
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When are internal waves nonhydrostatic?
CPU time 1 day CPU time 3 days
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Hydrostatic vs. Nonhydrostatic lock exchange
computation
Hydrostatic
Nonhydrostatic
Doman size 0.8 m by 0.1 m (grid 400 by 100)
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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.

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Speedup with the preconditionerwhen applied to a
domain with dD/L0.01
No preconditioner (22.8X)
Diagonal (8.5X)
Block-diagonal (1 X)
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Vertical grid structure z-level
SUNTANS employs z-level grids, which have the
advantage that they allow accurate computation
of horizontal density gradients static density
fields remain static.
Stair-stepped
Bottom-following
Advantage Stepped geometries
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Z-level prism grids with the Immersed boundary
method
The main drawback to using z-level grids is
accurate computation of Wall-normal gradients.
This is remedied with the use of the
ghost-cell immersed boundary method of Tseng and
Ferziger (2003).
Stair-stepped UW0 at boundaries
Immersed boundary U, W nonzero
Image Yi-Ju Chou, EFML
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Other features required when wetting and drying
is employed
Wetting and drying incurs small cell heights
Implicit vertical advection/diffusion
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Monterey Bay An internal wave sanctuary
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Internal tide generation over topography
Deep ocean (3000 m deep)
Shelf (500 m)
Depth
Density
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Internal wave generation in Monterey Bay
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Where are internal waves generated?
Energy flux
Across 1000 m of water,
E 1 MW
1 KW/m
Figure S. Jachec, EFML
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log10(W/m2)
52 MW
Figures S. Jachec, EFML
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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

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Internal Waves in the South China Sea

• How do internal waves influence the propagation
  of acoustic signals in the ocean?

Warm, light
Cold, heavy

• Numerical Challenges
• High aspect ratio 10 km long,
• 10 m high
• Long-time propagation 1 m/s
• over 3000 km
• Disparate length scales 10 km
• wavelength, 1 m interface

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Nonhydrostatic internal wave propagation
Hydrostatic
Nonhydrostatic
Small changes in wave character can significantly
alter propagation of acoustic signals
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Transport due to internal waves at Huntington
Beach
Initial tracer field
Without Stratification
Critical stratification
After 3 weeks
Supercritical stratification
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Acknowledgments

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