Issues regarding the use of hybrid coordinates

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Issues regarding the use of hybrid coordinates

• Rainer Bleck
• NASA/Goddard Institute for Space Studies
• August 2007

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• Principal design element of isopycnal models
  Depth and (potential) density trade places as
  dependent / independent variables
• - same number of unknowns, same number of
  (prognostic) equations, but very different
  numerical properties
• Driving force for isopycnal model development
  genetic diversity

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• Main benefits
• - explicit PV and potential enstrophy
  conservation
• - reduction of numerically induced diapycnal
  mixing during advection diffusion
• Main pitfalls
• - degeneracy in unstratified water column
• - 2-term horizontal PGF is error-prone in
  steeply inclined layers (reduction to 1 term
  possible at the price of approximating state
  eqn.)
• - layer outcropping (gt "massless" layers)
• - strongly varying layer thickness requires
  sophisticated advection schemes

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Grid degeneracy is main reason for introducing
hybrid vertical coordinate "Hybrid" means
different things to different people - linear
combination of 2 or more conventional coordinate
s (examples zsigma, zrho, zrhosigma) -
ALE (Arbitrary Lagrangian-Eulerian) coordinate
ALE maximizes size of isopycnic subdomain.
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ALE Arbitrary Lagrangian-Eulerian coordinate

• Original concept (Hirt et al., 1974) maintain
  Lagrangian character of coordinate but re-grid
  intermittently to keep grid points from fusing.
• In RUC and HYCOM, we apply ALE in the vertical
  only and re-grid for 2 reasons (1) to maintain
  minimum layer thickness (2) to nudge an
  entropy-related thermo- dynamic variable
  toward a prescribed layer-specific target
  value by importing water from above or
  below.
• Process (2) renders the grid quasi-isentropic

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MICOMs hybrid coordinate cousin HYCOM
c o o r d i n a t e l a y e r 1
c o o r -
d i n a t e
l a y e r 2
c o o r d i n a t e l a y e r 3
equator
Pole
Pole
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Montevideo
south
Vertical section through HYCOM solution. Heavy
black lines coordinate surfaces. Shaded
contours potential density
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Continuity equation in generalized (s)
coordinates
(zero in fixed grids)
(zero in material coord.)
(known)
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Present approach

• One-dimensional .. scheme is not allowed to look
  sideways.
• Present routine (hybgen) restores target
  densities exclusively by entraining water from
  neighboring layers
• Regridding is based on density profile. T S
  are subsequently remapped by 1st or 2nd order
  vertical advection scheme (PCM,PLM,PPM).
• Minimum layer thickness criteria have evolved
  over time to reduce shortcomings in vertical grid
  layout.

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1
Absorb near-massless layers on sea floor in
lowest substantial layer
Is lowest mass-containing layer too light?
no
Split into 2 sub-layers, one matching the target
density and one matching the density of the layer
above. Absorb upper sub-layer in layer above.
Maintain constant depth of top layer
Loop through remaining layers (top-down) to
restore target density
yes
Is lower intfc of layer k too close to surface?
Density of layer k different from target?
Too light
Too dense
Entrain water from layer k-1
Cant because intfc too close to srfc?
Transfer mass from k to k-1
No problem
Is lower intfc of layer k too close to srfc?
no
yes
Entrain water from layer k1
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Pressure force problems in HYCOM

• In idealized isopycnic models that disregard
  separate effects of T and S on compressibility,
  the pressure gradient is a single-term expression
  (involving M Fpa).
• Thermobaricity adds a second term to the pressure
  gradient expression. The added term can become
  large in steeply inclined coordinate layers.
• The magnitude of the added term depends on an
  arbitrarily chosen reference T/S profile.
• The choice of reference profile affects the
  modeled circulation.

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How most non-Cartesian models compute horizontal
pressure gradients
hydrostatic reduction
u
hydrostatic reduction
s1
s2
s3
s4
x1
x2
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Janjic 1 interprets the familiar pressure
gradient error in terrain-following (sigma)
coordinate models as resulting from violating the
requirement
a const over slab bounded by (x1,x2),
(p1,p2)
The same problem arises in HYCOM if isopycnic T/S
gradients occur in steeply inclined coordinate
layers.
1 Janjic, Z, 1977 Pressure gradient force and
advection scheme used for forecasting with steep
and small scale topography. Contrib. Atm. Phys.,
50, 186-199.
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More precise computation of horizontal pressure
gradients
u
s1
s2
hydrostatic integration
hydrostatic integration
s3
s4
x1
x2
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Finite volume approach
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The T/S/r conundrum in isopycnic-coordinate models

• T,S,rpot are materially conserved in adiabatic
  flow.
• The three variables are related rpot rpot(T,S)
  and T,S are of similar importance.
• Due to numerical errors and nonlinearities in the
  equation of state, T,S advection is unlikely to
  conserve rpot .
• This is extremely inconvenient in models using
  rpot as independent variable.

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l -0.13 psu/deg
l -0.26 psu/deg
Linearized spiciness c S l T
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r/spiciness advection works well, but T/S
advection may ultimately be the better strategy
due to its formal conservation properties
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Laplacian interface smoothing
Issue generation of negative layer thickness
values during interface smoothing
Biharmonic interface smoothing
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x
Laplacian interface smoothing
p
Biharmonic interface smoothing
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present 3-step procedure

• Convert interface pressure tendencies into layer
  thickness tendencies
• Construct preliminary bolus fluxes assuming
  each neighbor point contributes 25 to thickness
  tendency
• Limit bolus fluxes, using FCT technology, so as
  to avoid creating negative thicknesses

When computing 4th derivatives, do not use points
resting on elevated bottom topography
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Vertical T/S mixing in isopycnic subdomain

• Present approach KPP mixes T S grid generator
  then restores target density
• Suggested alternative use McDougall-Dewar
  algorithm originally developed for MICOM. It
  mixes T S (like KPP) but deduces layer
  thickness change required to keep density on
  target.

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r1
z
r2
r3
Perched water masses
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• The pinching off of layers by protruding bottom
  features inhibits draining of dense water
  generated in slope regions.
• Opening small gaps in pinched-off near-bottom
  layers has detrimental long-term effects on the
  circulation not yet fully understood.

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Remarks

• Sloping coordinate layers create problems not
  seen in z-coordinate models
• According to Spiegel Veronis (1960), it is OK
  to assess buoyancy effects on the basis of
  potential (as opposed to in-situ) density.
• In fact, present methods to compute PGF in
  isopycnic models do not work if layers are filled
  with water of in-situ density. Virtual potential
  density (Sun et al. 1999) works, sort of.
• More accurate methods (finite-volume based,
  Shchepetkin McWilliams 2003) for computing PGF
  may alleviate this problem, opening the door to
  the use of an unapproximated state eqn.

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Remarks (cont.)

• Rendering of ENSO in coupled models has been
  found to benefit from the presence of isopycnic
  coordinate layers near the surface (Sun Bleck,
  2006).
• Present philosophy in ALE models therefore has
  been to expand the isopycnic sub-domain by
• making the constant-thickness remnants of
  outcropped layers as thin as possible
• avoiding introduction of layers that by virtue of
  their low target density always default to
  constant thickness.
• The above strategy yields ridiculously degenerate
  grid layouts at high-latitudes
• Potential remedy spread out high-lat layers by
  assigning large minimum thickness values to
  densest coordinate layers (S.Sun, pers. comm.).

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Nino3 in GISS/HYCOM and
Observed Sensitivity to vertical discretizations
and mixed layer schemes

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MICOMs hybrid coordinate cousin HYCOM
c o o r d i n a t e l a y e r 1
c o o r -
d i n a t e
l a y e r 2
c o o r d i n a t e l a y e r 3
equator
Pole
Pole
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MICOMs hybrid coordinate cousin HYCOM
c o o r d i n a t e l a y e r 1
c o o r -
d i n a t e
l a y e r 2
c o o r d i n a t e l a y e r 3
equator
Pole
Pole
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Remarks (cont.)

• The road to layer modeling heaven is littered
  with discarded barotropic-baroclinic
  mode-splitting schemes.
• We are learning, so this issue should be off the
  table soon.