Nonhydrostatic effects of breaking interfacial waves on a sloping boundary

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Nonhydrostatic effects of breakinginterfacial
waves on a sloping boundary

• Oliver B. Fringer and Robert L. Street
• Environmental Fluid Mechanics Laboratory
• Stanford University
• 9/16/99
• Support DOE, Computational Science Graduate
  Fellowship

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Overview

• Introduction
• Model overview
• Nonhydrostatic (rigid lid)
• Quasihydrostatic (q0)
• Hydrostatic
• Test cases
• Surface seiche
• Interfacial wave dispersion
• Interfacial breaking waves
• Progressive waves
• Solitary waves
• Conclusions

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Internal Wave Basics
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1
2
1. Focusing 2. Overturns 3. Soliton Packets 4.
Lateral Transport

• Sediment Transport
• Acoustic Propagation
• Optical Clarity
• Deep ocean mixing

• When is a flow hydrostatic?
• In general H/L ltlt 1
• Continuous stratification U/NLltlt1, N buoyancy
  frequency
• Nonhydrostacy does not imply nonlinearity
• e.g. linear dispersive water waves

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Motivation

• Observations of large amplitude interfacial
  waves
• Highly nonlinear solitons observed off of Oregon
  coast 25 m amplitude in 30 m depth (Stanton and
  Ovstrosky, 1998)
• High amplitude isopycnal displacements in
  Monterey Bay, CA 120 m in 220 m depth (Paduan
  and Rosenfeld, 1997)
• Intermittency and sparsity of internal wave
  breaking makes them very difficult to measure,
  yet they generate a significant portion of
  dissipation within the ocean.
• Linkage with laboratory scale study provides
  strong support for model verification.

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Goals
Determine the nonhydrostatic effects associated
with interfacial wave breaking
Using LES in conjunction with laboratory
experiments, determine the 2 and 3-dimensional
dissipative and mixing mechanisms associated with
interfacial wave breaking
Parameterize the nonhydrostatic effects to be
used in hydrostatic models.
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Model Overview
3D Navier Stokes Nonhydrostatic (Zang,Street 93)
TRIM Quasihydrostatic (Casulli,Stelling 96)

• Finite volume
• Generalized curvilinear grid
• Nonstaggered variables
• Quick/Sharp for advection
• Rigid lid
• Fractional step
• Multigrid

• Rectilinear grid
• Staggered variables
• Euler-Lagrange for advection
• Semi-implicit free surface
• Nonhydrostatic corrector
• Conjugate gradient

3D Navier Stokes Quasihydrostatic

• Finite volume
• Restricted curvilinear grid
• Nonstaggered variables
• Quick/Sharp for advection
• Semi-implicit free surface
• Quasihydrostatic corrector
• Multigrid

3D Mesoscale approximation (Mahadevan, et. al.
96)
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• Restricted free surface following curvilinear
  grid
• Horizontal metrics independent of depth
• Nonstaggered variables

only need to recompute these metrics at each time
step
Top View
Side View
contravariant volume flux
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• Curvilinear free surface equation advanced
  semi-implicitly
• Pressure splitting
• Curvilinear momentum equation
• 3D Poisson Equation
• 2D Poisson Equation

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• Method properties
• Local and global volume conservation
• No free surface stability restriction
• Stability limitation is governed by viscosity at
  boundaries, because resctricted coordinate
  viscous term cannot be made implicit
  efficiently
• Pressure is first order accurate in time at best
  for any projection method (Armfield, 1999)
• Scheme is first order acurate in time resulting
  from nonstaggered pressure discretization error
  (Armfield and Street, 1999), hence it is stable
  for theta .5 because the pressure error
  introduces dissipation.
• q0 free surface boundary condition inhibits
  convergence for flows which are closeley
  hydrostatic with sparse regions of nonhydrostasy.
• Generalized curvilinear coordinates introduce
  false baroclinic gradients when grid is angled
  with respect to density profile (e.g. POM)

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• Pressure heirarchy
• Fully nonhydrostatic (rigid lid)
• Quasihydrostatic (q0 boundary condition)
• Hydrostatic

Benefits -Solution of one 19 diag. -Solves for
all pressure. Drawbacks -No free surface dynamics
Benefits -Solves for free surface. -No free
surface stability lim. Drawbacks -one 19 diag
and one 9 diag.
Benefits -Cheap. Drawbacks -poor for slight
nonhydrostacy -ill posed at boundaries
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Test Cases

• Free surface wave dispersion
• test linearized water wave theory
• linearized dispersion relation c2 g
  tanh(kH)/k
• c2 g H
• (Hydrostatic, nondispersive shallow water
  limit, kH-gt0)
• c2 g / k
• (Nonhydrostatic, deep water limit, kH-gt8)

Initial condition
H
2d Grid 20x20, CFL.1 vary H, constant k2p/L
L
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(No Transcript)
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• Linear surface wave results

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• Interfacial wave dispersion (modeling a
  wavemaker)
• tanh salinity profile ?s4.28 ppt, thickness1
  cm, depth60 cm, Nmax.8Hz
• Vary frequency w, keep L 5l
• grid 2D 128x64, ?t5T/2000, ?z(min) 2 mm
• Miscible interface
• Trouble forcing primitive equations since
  hydrostatic solution is the w 0 case.

c
free surface
ss-?s/2
open boundary (Javam and Armfield, 1997)
Specify u, w ?s/?x0
ss?s/2
free slip
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• Forcing interfacial waves
• Apply linearized Euler-Boussinesq
  equations
• Mode shapes for waves of frequency 0.1 Hz

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• Testing the code with the dominant first mode
  waves
• Determine the range of frequencies (0ltwltN) of
  interest.
• Solve the aformentioned Sturme-Liouville problem
  and determine the first mode shape and its speed
  (Eigenvalue solver from Num. Rec. in Fortran)
• Force the Quasihydrostatic code with u and w for
  5 periods.
• Measure the wavelength and the response frequency
  with FFTs (Matlab).

Dispersive behavior is very robust!
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• Short Waves
• High frequency
• Nonhydrostatic
• Less dispersive

• Long Waves
• Low frequency
• More hydrostatic
• more dispersive

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Interfacial Breaking Waves

• Progressive wave (Experiment of Troy and Koseff,
  EFML)
• Simulation parameters
• Wave type Progressive forced with .1 Hz linear
  interfacial wave.
• w .1 Hz, l 1.8 m
• Grid 128 x 32 x 64, ?t .015 s, CFL.5 based on
  Umax, tmax90 sec (6000 steps)

Boundary conditions free slip everywhere
but free surface q0 BC and left forcing boundary.
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2D Simulation

• Phase 1 2D nonlinear steepening
• The initial instability is 2D and results from
  the relatively long wave encountering the slope
  as the leading trough moves down the slope.
• Phase 2 3D Longitudinal Rolls /Rayleigh-Taylor
  billows
• Wave crest overtakes the trough, forcing heavy
  fluid over light fluid.
• Leading wave trough generates an intense shear
  layer which leads to a Couette instability.
• Phase 3 Kelvin-Helmholtz Billows.

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• Demonstration of progressive wave longitudinal
  vorticity production at t36.0 s RED SURFACE
  ??1rad/sec BLUE SURFACE ??-1rad/sec

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Progressive .1 Hz s0 salinity isosurface
breaking on a slope.
t10.5 s
t0.375 s
overturning
shoaling
receding
t21.0 s
t28.5 s
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Rayleigh-Taylor billows
longitudinal rolls
Kelvin-Helmholtz
t34.125 s
t36.750 s
t41.625 s
t59.250 s
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• Solitary wave (Experiment of Michallet and Ivey,
  1999)
• Begin with an initial Gaussian depression of
  depth .15 m below the 1 cm thick interface and
  width .58 meters. Wave propagates at the shallow
  water speed of .26 m/s towards the 7.25
  slopeSimulation parameters
• Wave type Solitary
• Grid 128 x 32 x 64, ?t .005 s, CFL.12 based
  on Umax, tmax45 sec (9000 steps)
• Initial conditions
• Initial potential energy is given by (assuming
  sharp interface)

Boundary conditions free slip everywhere
but free surface q0 BC
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2D case for tmax40 sec, showing overturning and
KH instability
Magnified free surface (100h) at t0
t30 s
t32 s
t35 s
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• Solitary Wave Energy budgets
• Total energy is given by
• Normalized energy
• Determine Amount of energy dissipated during
  breaking event.
• Runs performed
• 2/3D nonhydrostatic
• 2/3D quasihydrostatic
• 2/3D hydrostatic

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t18.2 s.
t 25.2 s.
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• Comparison of energy budgets for a solitary wave
  breaking on a slope

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• Comparison of energy dissipation for a solitary
  wave breaking on a slope.

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• Energy budgets for 2 and 3d hydrostatic breaking

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Conclusions

• q0 free surface quasihydrostatic pressure
  boundary condition inhibits quasihydrostatic
  multigrid convergence.
• Interfacial wave dispersive behavior is highly
  robust.
• Interfacial wave breaking simulations
  qualitatively similar to experiments.
• Hydrostatic model does not generate a breaking
  instability and adds energy to solitary wave
  because vertical momentum is overestimated.
• Quasihydrostatic model mimics nonhydrostatic
  dissipation but overshoots energy during
  steepening event
• q0 quasihydrostatic boundary condition inhibits
  complete nonhydrostatic dynamics and weakens
  overturning strength due to slower horizontal
  velocity at the wave crest.
• Maximum dissipation occurs after overturn as wave
  moves upslope and maximizes strength of
  longitudinal vorticity.
• dissipative source is numerical, resulting from
  the high cross stream shear due to this vorticity
• 2D simulations cannot dissipate energy via cross
  stream shear because they lack longitudinal
  vorticity resulting from the 3D instabilities.
• Steepening mechanism is stronger for the
  nonhydrostatic case, so this increased shear
  generates the increased dissipation for the 2D
  nonhydrostatic case.