Comparison of Vertical Mixing Parameterizations for the WindDriven Coastal Ocean

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Comparison of Vertical Mixing Parameterizations
for the Wind-Driven Coastal Ocean
Scott Durski Oregon State University
Thanks to Dale Haidvogel, Scott Glenn, Hernan
Arango and John Allen
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Large, McWilliams and Doney, K-Profile
parameterization
Boundary layer mixing
Interior mixing
Mixing parameterized as a function of boundary
layer depth, turbulent velocity scale and a
dimensionless shape function.
Shear generated mixing
Based on gradient Richardson Number
Ko
Boundary layer depth
Based on bulk Richardson number.
0.5
Rig
Internal wave mixing
Constant background mixing value from open
ocean thermocline observations.
Turbulent velocity scale
From atmospheric surface boundary layer
similarity theory
Double diffusive mixing
Based on laboratory measurements, based on
density ratio.
Shape function
Third order polynomial with coefficients
determined from boundary conditions at surface
and ocean interior.
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Mellor-Yamada Level 2.5 turbulence closure scheme
Vertical mixing is estimated based on two
prognostically calculated quantities, tubulent
kinetic enery and a turbulent length scale.
Background mixing coefficients
TKE equation
q2l equation
Turbulent quantities are advected and diffused
both vertically and horizontally. Shear
production of TKE occurs at a flux Richardson
Number below 0.19. Wall proximity function W
enforces surface layer similarity theory at
surface and bottom boundaries.
(from Kantha and Clayson, 1994)
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1-D wind driven boundary layer test case
Circulation model
ROMS
(with a single tracer equation to represent
density rather than separate T and S equations.)
Setup
Double-periodic uniform domain to represent
one-dimensionality. 25 meter water depth
discretized into 80 evenly spaced levels.
Initialization
At rest. Well-mixed water column down to 7.5
meters depth Uniform N2 below this depth.
Forcing
0.03 N/m2 wind stress (spun-up over half a day)
Test cases
Stratification below the mixed layer of 4
different intensities to represent the range
observed in the coastal ocean.
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Time-series of surface density for the four cases

• KPP entrains more at lower stratification than
  M-Y.
• M-Y entrains more at very high stratification.
• Entrainment is more intermittent with KPP.

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Features of the circulation
No/10 case - KPP
density
Stratification intensifies at base of boundary
layer. Inertial oscillations develop causing
periods of enhanced and reduced shear at the
pycnocline. Stratification intensifies and
relaxes out-of-phase with shear (particularly
strongly with KPP).
v-velocity
u-velocity
N2
velocity shear
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Vertical flux divergence
KPP shows much stronger response to incidents of
enhanced shear at base of b.l, while exhibiting
less entrainment at other times. Flux is
alternately positive and negative at pycnocline
with KPP, but not so with M-Y.
KPP
(No/10 case)
M-Y
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Onset of turbulence at the pycnocline
Turbulence retreats upwards with KPP during low
shear periods. M-Y maintains mixing in more
highly stratified waters at high Ri during these
periods as a result of diffusion and finite-rate
of TKE dissipation. During high shear periods KPP
penetrates more deeply into the stratification at
a higher Richardson number.
KPP
M-Y
Konset 5 x 10-6
Konset
Rig at Konset
N2 at Konset
Shading indicates periods of low shear at base of
boundary layer.
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Why does KPP entrain more in low stratification
and M-Y entrain more in high stratification?
Which scheme mixes more depends on whether
sustained entrainment during low shear periods by
M-Y predominates enhanced mixing during high
shear periods with KPP.
2No
No/10
Higher stratification at base of boundary layer
does not lead to equally higher shear there.
(Shear is primarily controlled by the surface
stress in this case.) Consequently depth of
penetration of shear-generated-mixing is greatly
reduced at high N2.
vertical profile of N2
maximum velocity shear2
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Modifications to the K-profile Parameterization
for coastal ocean application.
The original KPP scheme must be appended with a
bottom boundary layer approximation to avoid
erroneously high estimates of mixing
(particularly in shallow water and near the
bottom).
Profiles of turbulent mixing coefficient in
different shallow water situations
?
?
?
?
?
1) Matching with log layer similarity theory
where surface boundary layer extends to the
bottom..
3) Apply matching rules when surface and bottom
boundary layers overlap. (BBL estimate
over-writes SBL estimate where they overlap)
2) Add a K-profile parameterization for the
bottom boundary layer modeled after SBL scheme.
Note Only a neutral bottom boundary layer is
considered.
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Interacting boundary layers, 1-D model setup.
Setup
Double-periodic uniform domain. 15.5m, 16.0m,
18.0m and 20m water depth discretized into 80
evenly spaced levels.
Initialization
At rest. Well-mixed water column down to 7.5
meters depth Uniform N2 within a pycnocline of
thickness 0.5m, 1m, 3m and 5m
Forcing
0.03 N/m2 wind stress and oppositely directed
bottom stress (spun-up over half a day)
Test cases
Vary both intensity of stratification in
pycnocline and thickness of the initial
pycnocline.
0.03 N/m2
h
N2
-0.03 N/m2
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Variation in entrainment with pycnocline
thickness for high stratification case.
Shear-generated mixing within the pycnocline
between the boundary layers leads to enhancement
of mixing with KPP for suitably thin interfaces.
Time-series of surface density
(No case)
KPP
M-Y
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Evolution of the pycnocline with lower
stratification
A transition occurs between maintaining an
intensified pycnocline when the initial thickness
of the stratified layer is high, to the rapid
development of a well-mixed water column when it
is thin.
(No/10 case)
Time-series contour plots of density over the
water column
Thickness of pycnocline
KPP
MY
1 m
3 m
5 m
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Divergent response with the two schemes for a
1.5m thick pycnocline
The response with the two schemes is markedly
different when the initial pycnocline thickness
is 1.5 meters.
Time-series contour plots of density over the
water column
KPP
MY
Time-series contour plots of v-velocity over
the water column
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Interacting boundary layers 2-D coastal
upwelling on a shallow shelf
Setup
250 meter resolution in the across-shore
direction. Periodic in the alongshore direction.
40 evenly spaced vertical levels. 6 meter depth
at coast, sloping downward offshore at a rate of
1 m/km
Initialization
At rest. Well-mixed water column down to 7.5
meters depth Uniform N2 within a pycnocline of
thickness 1m, 3m and 5m
Forcing
0.03 N/m2 wind stress (spun-up over half a day)
Test cases
Vary

• intensity of stratification in pycnocline,
• thickness of the initial pycnocline
• bottom slope.

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Upwelling with a highly stratified pycnocline
For strongly stratified pycnoclines, advection
dominates vertical mixing in the evolution of the
two-dimensional upwelling.
(No, 1m thick initial pycnocline)
KPP
M-Y
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Upwelling at low stratification
Vertical mixing competes with advection at
lower stratification
KPP forms a stronger surface-upwelling-front
earlier due to stronger mixing where surface and
bottom boundary layers interact. Early offshore
migration of upwelling front is dictated by
vertical mixing, not Ekman transport with KPP.
(No/20, 5 m thick initial pycnocline)
Day 0.75
Day 1.75
Day 3.0
Day 4.5
KPP
M-Y
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Impact of choice of vertical mixing
parameterization on a simple biological model.
Consider a simple model of an organism that can
obtain a bloom state if a large enough
seed-population co-occurs with adequate food
supply. Given initial presence of organisms
only at or above the pycnocline and abundance of
food below the pycnocline, what develops with
the two vertical mixing schemes?
A significant bloom of the organism develops in
the near-shore region with KPP, while
concentrations remain low everywhere with M-Y.
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Conclusions
KPP responds more strongly to shear mixing events
in the interior. M-Y sustains mixing in
intermittently high Rig environments via
diffusion of TKE and due to the finite rate of
decay of turbulence in the model. The enhancement
of mixing due to shear with KPP becomes less
potent ad the stratification of the pycnocline
increases. KPP generates more mixing in the
interior of the pycnocline, where surface and
bottom boundary layers interact. Very distinct
differences in the resultant circulation arise
for particular initial conditions which may cause
the results obtained from the two schemes to
diverge in more complicated (realistic)
settings. Even when the differences in the
physical fields produced with the two schemes are
small, systems which respond non-linearly to
these differences may exhibit dramatic
differences.