Contradictory Considerations in Choosing A Vertical Coordinate

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Contradictory Considerations in Choosing A
Vertical Coordinate

• THE VERTICAL COORDINATE MUST BE MONOTONIC WITH
  DEPTH FOR ANY PHYSICALLY ATTAINABLE STATE OF
  INTEREST.
• THE SOLENOIDAL PRESSURE GRADIENT TERM SHOULD BE
  RELATIVELY SMALL COMPARED TO THE NON-SOLENOIDAL
  PRESSURE GRADIENT TERM WITH AN ACCURATE EQUATION
  OF STATE
• MATERIAL CHANGES IN TEMPERATURE SALINITY DUE TO
  NUMERICS SHOULD BE MUCH SMALLER THAN CHANGES DUE
  TO PHYSICAL PROCESSES.
• IT SHOULD BE POSSIBLE TO CONCENTRATE RESOLUTION
  WHEREVER IMPORTANT PROCESSES OCCUR, INCUDING
  BOUNDARY LAYERS AND INTERIOR REGIONS OF LARGE
  GRADIENTS.
• COORDINATE SURFACES THAT APPROXIMATE
  LOCALLY-REFERENCED NEUTRAL SURFACES PERMIT A
  NEARLY TWO-DIMENSIONAL REPRESENTATION OF
  ADVECTION AND ISONEUTRAL MIXING.
• CONSISTENCY IS MUCH EASIER TO ESTABLISH WITH A
  SINGLE VERTICAL COORDINATE.
• THE COORDINATE SHOULD MAKE THE TOP AND BOTTOM
  BOUNDARY CONDITIONS EASY TO IMPLEMENT EXACTLY.
• THE COORDINATE SHOULD FACILITATE ANALYSIS OF
  SIMULATIONS.

There is no right answer for all applications!
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• Key questions in evaluating an ocean climate
  model
• Is the near-surface thermodynamics adequately
  represented?
• Are watermass formation processes represented
  correctly?
• Are watermass destruction processes represented
  correctly?
• Do marginal seas have the right processes
  governing their properties?
• Are the currents sensible?
• Will the results degrade with climate change?
• Issues related to the choice of vertical
  coordinates
• How well are the mixed-layer and its interaction
  with the interior represented? (r)
• How well are gravity currents represented? (Z, s)
• Is there too much numerically induced diapycnal
  mixing? (Z, s)
• Is the equation of state correctly represented in
  the dynamics? (r)
• Has smoothing of topography grossly altered the
  circulation? (s)
• Is the current state of the ocean somehow
  embedded in the model? (Z, s, r)

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Primitive Equation Models and Vertical Coordinates

• Nonhydrostatic Non-Boussinesq (Navier-Stokes)
  Equations
• Hydrostatic Non-Boussinesq Equations with
  Traditional Approximation

3-D Momentum - Predictive equation for velocities
Continuity - Predictive equation for density
Tracer concentration - Predictive equation for
state variables
State - Diagnostic equation for pressure
2-D Momentum - Predictive for horizontal
velocities
Hydrostatic (vertical momentum) - Predictive for
pressure
Continuity Diagnostic equation for w
Tracer concentration - Predictive equation for
state variables
State - Diagnostic equation for density
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Hydrostatic Primitive Equations in Arbitrary
Coordinates
Horizontal Momentum
- or -
Hydrostatic
Continuity
Tracer conservation
State
No normal flow boundary condition
Free surface boundary condition
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Z-, sigma- and density-coordinates
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Hydrostatic Boussinesq Primitive Equations in
Geopotential Coordinates
Horizontal Momentum
Hydrostatic
Continuity
Tracer conservation
State
No normal flow boundary condition
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Strengths and Weaknesses of Geopotential (Z)
Coordinate Models

• Strengths
• Simple form of equations
• Simple form of pressure gradient term no
  baroclinicity in pressure gradient errors
• Most intuitive analysis for non-oceanographers
• Extensive experience in climate studies
  surprises are unlikely.
• Traditional Weaknesses
• Eddy-rich simulations exhibit very large spurious
  diapycnal mixing. (Griffies et al., 2000)
• (Addressed more fully by S. Griffies, next.)
• Overflows BBL representation very poor or
  complicated unless both horizontal and vertical
  resolution are high enough to resolve the BBL.
  (Winton et al., 1998)
• Downslope diffusion or
• Plumbing schemes help somewhat
• (e.g., Beckmann Doscher, 1993
• Campin Goose, 1999
• Killworth Edwards, 1999)
• Rigid lid surface boundary condition fits best
  with a pure Z-coordinate formalism.
• Near-surface resolution limited by
  surface-height/sea-ice displacement.
• Completely avoided by using stretched Z instead
  of Z. (Adcroft and Campin, 2004)
• Poor representation of topography? (e.g. Gerdes,
  1993)

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Z as an Improvement on Z
Z-coordinate
s-coordinate
Z-coordinate
(Adcroft and Campin, 2004)
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Hydrostatic Primitive Equations in Pressure
Coordinates
Horizontal Momentum
Hydrostatic
Continuity
Tracer conservation
State
No normal flow boundary condition
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Strengths and Weaknesses of Pressure Coordinate
Models

• (Very similar to Geopotential Coordinate models,
  but more natural when the Boussinesq
  approximation is not made)
• Traditional Strengths
• Simple form of equations
• Simple form of pressure gradient term no
  baroclinicity in pressure gradient error.
• Straightforward Analysis
• Direct prediction of sea-surface height changes
  (because non-Boussinesq).
• Traditional Weaknesses
• May have large numerical diapycnal mixing from
  advection.
• Overflows are difficult to represent without high
  horizontal and vertical resolution.
• Difficult to enhance resolution near bottom.
• Near-bottom resolution must be less than bottom
  pressure variations.
• Switching from P to P ameliorates this.
• Complicated bottom boundary condition

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Hydrostatic Primitive Equations in Terrain
Following Coordinates
(For example)
Horizontal Momentum
Hydrostatic
Continuity
Tracer conservation
State
No normal flow boundary conditions
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Strengths and Weaknesses of Terrain-following
Coordinate Models

• (Issues for global climate application addressed
  in detail by G. Danabasoglu later.)
• Strengths
• Topography is represented very simply and
  accurately
• Easy to enhance resolution near surface.
• Lots of experience with atmospheric modeling to
  draw upon.
• Traditional Weaknesses
• Pressure gradient errors are a persistent
  problem.
• Errors are reduced with better numerics (e.g.,
  Shchepetkin McWilliams, 2003)
• Gentle slopes (smoothed topography) must be used
  for consistency
• Traditional requirement for stability (Beckman
  Haidvogel, 1993)
• ROMS requirement (Shchepetkin, pers. comm)
• Spurious diapycnal mixing due to advection may be
  very large. (Same issue as Z-coord.)
• Diffusion tensors may be especially difficult to
  rotate into the neutral direction.
• Strongly slopes require larger vertical stencil
  for the isoneutral-diffusion operator.

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Hydrostatic Primitive Equations in Isopycnal
Coordinates
Potential Density
Layer thickness
Specific Volume
Montgomery potential
Horizontal Momentum
Hydrostatic
Continuity
Tracer conservation
Diapycnal velocity
State
Boundary conditions
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Strengths of Isopycnal Coordinate Ocean Models

• EXACT DISCRETIZATION OF A PHYSICALLY REALIZABLE
  SYSTEM.
• Close analog to watermass description of flow.
• Close analog to theoretical/analytic models.
• There is an exactly conserved Potential Vorticity
  analog
• THERE IS NO NUMERICAL DIAPYCNAL DIFFUSION.
• RESOLUTION MIGRATES TO REGIONS OF HIGH
  STRATIFICATION.
• Gravity currents strong thermal wind shears
  tend to be naturally well resolved.
• THERE IS NO SOLENOIDAL PRESSURE FORCE TERM.
• INTERNAL GRAVITY WAVES DO NOT REQUIRE ADVECTIVE
  TERMS.
• Very natural split between dynamic and
  thermodynamic equations
• With many tracers, high resolution isopycnal
  models can be relatively fast.
• CAN BE USED WITH ARBITRARY TOPOGRAPHY.
• E.g., HIM is the basis of the most accurate
  published forward (non-assimilative) global tide
  model. (Arbic et al., 2004)
• ADVECTION AND DIFFUSION ARE PRIMARILY
  TWO-DIMENSIONAL.
• Diffusion tensor can follow coordinate surfaces
  (?)

Properties that do not hold exactly with a
nonlinear equation of state.
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Traditional Weaknesses of Isopycnal Coordinate
Ocean Models

• NONLINEARITIES OF THE EQUATION OF STATE ADD
  SIGNIFICANT COMPLEXITY.
• (Described in detail later by R. Hallberg.)
• Most of these issues are now solved, after 10
  years of work.
• THERE IS NO MATERIALLY CONSERVED QUANTITY THAT IS
  EVERYWHERE MONOTONIC WITH DEPTH. (INSOLUBLE)
• RESOLUTION IS NATURALLY EXCLUDED FROM
  UNSTRATIFIED REGIONS, INCLUDING THE PLANETARY
  BOUNDARY LAYER.
• Ameliorated by coupling to a separate bulk or
  refined-bulk mixed layer model.
• NUMERICS TEND TO BE MORE COMPLICATED THAN THE
  SIMPLEST AVAILABLE WITH Z-COORDINATES.
• Equations have nonlinear thickness dependencies
  in terms that are otherwise linear.
• Numerics must accommodate vanishing layers.
• No credible climate models use the simplest
  Z-coordinate numerics anyway.
• DENSITY INVERSIONS CAN NOT BE REPRESENTED.
• Isopycnal coordinates have limited utility in
  nonhydrostatic models.
• NON-OCEANOGRAPHERS SEEM TO GET VERY CONFUSED BY
  ISOPYCNAL COORDINATE MODELS.

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Pacific Density Surfaces from an Isopycnal Ocean
Climate Model
February Interfaces from a 48-layer, 1 Global
Isopycnal Model along 140W
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The Challenge of Hybrid Vertical Coordinate Models

• (To be discussed by R. Bleck later.)
• How does one design of a Hybrid vertical
  coordinate model that capture the best aspects of
  the coordinates it emulates, and avoids the
  worst?
• How well are we doing so far?

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Remaining Talks in Session

• Spurious Diapycnal Mixing In Ocean Models
  (Stephen Griffies, GFDL)
• Issues Arising From The Nonlinear Equation Of
  State In Isopycnal Coordinate Models (Robert
  Hallberg, GFDL)
• Are There Remaining Issues Precluding The Use Of
  Terrain-following Coordinates In Ocean Climate
  Models? (Gokhan Danabasoglu, NCAR)
• Issues Regarding The Use Of Hybrid Coordinates
  What Considerations Give The Best Of The Various
  Coordinate Options, And Not The Worst? (Rainer
  Bleck, GISS)
• Discussion

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The following slides are optional, following up
on the question of how readily overflows can be
represented.
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Resolution requirements for avoiding numerical
entrainment in descending gravity currents.

• Z-coordinate
• Require that
• AND
• to avoid numerical entrainment.
• (Winton, et al., JPO 1998)
• Many suggested solutions for Z-coordinate models
  
• "Plumbing" parameterization of downslope flow
• Beckman Doscher (JPO 1997), Campin Goose
  (Tellus 1999).
• Adding a separate, resolved, terrain-following
  boundary layer
• Gnanadesikan (1998), Killworth Edwards (JPO
  1999), Song Chao (JAOT 2000).
• Add a nested high-resolution model in key
  locations?
• Sigma-coordinate Avoiding entrainment requires
  that
• But hydrostatic consistency requires
• Isopycnal-coordinate Numerical entrainment is
  not an issue - BUT
• If resolution is inadequate, no entrainment can
  occur. Need

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Horizontal Resolution (in km) Required to Permit
50m Vertical Resolution at Bottom
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Horizontal Resolution (in km) Required to Permit
50m Vertical Resolution at Bottom
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Horizontal Resolution (in km) Required to Permit
50m Vertical Resolution at Bottom