Parameterizations of Resolved-Flow Driven Mixing and Planetary Boundary Layers
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Transcript of Parameterizations of Resolved-Flow Driven Mixing and Planetary Boundary Layers
Parameterizations of Resolved-Flow Driven Mixing and Planetary Boundary Layers
What is stratified shear mixing?• When vertical shears of velocity are large enough, enough kinetic energy
can be released by mixing to overcome the potential energy increase due to mixing against a density stratification, and mixing can spontaneously arise.
• The necessary condition for instability is given by the shear Richardson number: StableRiRi Critg 25.0 2
2
20/
S
N
u
gRi
z
zg
Simulated Kelvin-Helmholtz instability
z (m
)
x
Tem
pera
ture
(°C
)
u (m s-1)
0
0.02-0.02
0.5
-0.5 0
Where does stratified shear mixing matter in the ocean?
1. Dense overflows• Most interior ocean watermasses form through dense
overflows.2. Abyssal cataracts3. Equatorial Undercurrent
• The equatorial current and density structure are critical for ENSO.
4. Base of the surface mixed-layer• Property fluxes into the interior through non-deepening mixed
layers may are important.5. Wherever internal gravity waves steepen and break (maybe).
• Critical slopes?• Parametric Subharmonic Instability (PSI)?
All of these regions are important in the performance of large-scale ocean models, and need to be parameterized.
The parameterization of resolved-flow driven mixing must be the same for all regions!
Observed profiles from Red Sea plume from RedSOX (Peters and Johns, 2005)
Well-mixedbottom boundarylayer
(see Legg et al. 2006)
Actively mixinginterfacial layer
Shear param.appropriate here.
Shear-driven mixing of stratified turbulence
Abyssal Overflows – the Romanche Fracture Zone
Potential Temperature along Romanche Fracture Zone
Ferron et al., JPO 1998
Climatological Potential Temperature at 5000 m Depth
Equatorial Undercurrent Shear Mixing
Wind stress
Eastward Equatorial Undercurrent
Westward Current in Surface Mixed Layer
Isotherms
Side view along the equator
EasternPacific
WesternPacific
Impact of Shear-Mixing Parameterization on the EUC
Annual Mean Pacific EUCRicrit = 0.2 and Eo = 0.005
June Pacific EUC with Ricrit = 0.8 and Eo = 0.1 (Original values)
Shear-mixing at the base of the mixed layer
Density
Dep
th
Velocity
MechanicalStirring
Seasurface
Wind Stress &Turbulent Stress
Shearmixing?
Stratified shear mixing at the base of the surface mixed layer figures prominently in such idealized mixed layer models as Pollard, Rhines & Thompson (1973) or Price, Weller & Pinkel (1986)
Shear-related mixing due to internal waves hitting a slope (Sonya Legg, Princeton U.)
Failure and Success of Existing Shear Mixing Parameterizations• A universal parameterization can have no dimensional “constants”.
• KPP’s interior shear mixing (Large et al., 1994) and Pacanowski and Philander (1982) both use dimensional diffusivities.
• The same parameterization should work for all significant shear-mixing.• In GFDL’s GOLD-based coupled model, Hallberg (2000) gives too much mixing in
the Pacific Equatorial Undercurrent or too little in the plumes with the same settings.• To be affordable in climate models, must accommodate time steps of hours.
• Longer than the evolution of turbulence.• Longer than the timescale for turbulence to alter its environment.
• Existing 2-equation (e.g. Mellor-Yamada, k-, or k-) closure models may be adequate.• The TKE equations are well-understood, but the second equation (length-scale, TKE
dissipation, or TKE dissipation rate) tend to be ad-hoc (but fitted to observations)• Need to solve the vertical columns implicitly / iteratively in time for:
1. TKE2. TKE dissipation / TKE dissipation rate / length-scale3. Stratification (T & S)4. (and 5.) Shear (u & v)
• Simpler sets of equations may be preferable.• Many use boundary-layer length scales (e.g. Mellor-Yamada) and are not obviously
appropriate for interior shear instability.
However, sensible results are often obtained by any scheme that mixes rapidly until the Richardson number exceeds some critical value. (e.g., Yu and Schopf, 1997)
3
2
22
)1,7.0
min(1005.0
Ri
s
mRi
(At least) two equations for dimensional quantities are needed to describe turbulent mixing generically.
One is the turbulent kinetic energy per unit mass (TKE) equation:
With the usual Fickian (diffusive) closure it is
And with small aspect ratio
There are many options for the second equation, all very empirical:
Two-equation turbulence closures
w
g
x
UuuTransport
Dt
DQ
j
iji
0
233 NUUQDt
DQ zUQ
2
2
Nz
U
z
Q
zt
Q zzU
zQ
221
iuQ
2/1
23 ][
Q
Ql
smQl
XX
2
32 ][
Q
sm
XX
Q
sQ
XX
][/ 1
Mellor-Yamada 2.5
TKE-dissipation (k-)
TKE-dissipation rate (k-)
≡ Dissipation of Q
The second equation – e.g. TKE & dissipation (k-)
• TKE dissipation is (in)directly measurable.
• None of the terms in a dissipation equation are measurable.
• The functional form is chosen to mimic the TKE equation itself, with plenty of empirical constants added, mostly using boundary layer data.
UNS
NSUUQ
Q
,Pr
1
,2
22 NSz
Q
zDt
DQUQ
92.1744.044.1
2.122
NSQzzDt
DU
Q
2
22
2
22
,
QNQSNS
SN
SNNS
SNSN
SNNSU
mmm
nnn
ddddd
nnn
210
210
24
23210
210
,Pr
,
The n# m#, and d# are empirical constants.
See Umlauf & Burchard (Cont. Shelf Res., 2005) for a review.
≡ Dissipation of Q
221
iuQ
Properties:• Simple enough to solve iteratively along with its impacts.• Complete enough to capture the essence of stratified shear instability.
• Uses a length scale which is a combination of the width of the low Ri region (where F(Ri)>0), the buoyancy length scale LBuoy = Q1/2/N, and the distance from the boundary z-D.
• Decays exponentially away from low Ri region.
• Vertically uniform, unbounded limit:
• Ellison and Turner limit (large Q): reduces to form similar to ET parameterization
• Unstratified limit: similar to law-of-the-wall theories
Entrainment-law Derived Parameterization for Shear-driven Mixing (L. Jackson, R. Hallberg, & S. Legg, JPO 2008)
2BuoySL
[m2 s-1] Shear-driven diapycnal diffusivity / viscosity (Assumes Prandtl Number = 1)
Q [m2 s-2] Turbulent kinetic energy per unit mass
N2 = -g/ ∂/∂z [s-2] Buoyancy Frequency
, cN, cS [ ] Dimensionless (hopefully universal) constants
RiFz
u
Lz
2
222
2
DzNQL ,/min 2/1
At boundaries: Diffusivity: = 0 TKE/mass: Q = Q0 (≈ 0)
Dt
DQQ
z
ucNcN
z
u
z
Q
z SN 021
2222
2
0
0 0.1 0.2 0.3 0.4
0.03
0.12
0.06
0.09
0
Ri
F(R
i)
Simulated Shear-Driven Mixing
Kelvin-Helmholtz instability
3D stratified turbulence
z
x z
x
Shear results
DNS data
ET parameterisation JHL parameterisation
RiCr = 0.25, cN = 0.30, cS = 0.11, = 0.85
RiCr = 0.30, cN = 0.25, cS = 0.11, = 0.79
RiCr = 0.35, cN = 0.24, cS = 0.12, = 0.80
Jet results
DNS data
ET parameterisation JHL parameterisation
RiCr = 0.25, cN = 0.30, cS = 0.11, = 0.85
RiCr = 0.30, cN = 0.25, cS = 0.11, = 0.79
RiCr = 0.35, cN = 0.24, cS = 0.12, = 0.80
Shear results
Jet results
Comparison to other two-equation turbulence models
JHL
Important Processes for Determining Mixed Layer Depth
• Wind stirring-driven entrainment• Conversion of resolved shears to
small-scale turbulence (Richardson number criteria)
• Convective deepening• Overshooting convective plumes• Buoyancy-forced retreat to a Monin-
Obuhkov depth• Penetrating shortwave radiation• Vertical decay of TKE
• Restratification due to ageostrophic shears in mixed layer (Ekman-driven, eddy-driven, and viscous stresses on thermal wind shears)
Mostly at the base of the mixed layer and in the underlying transition layer.
A “refined” bulk mixed layer, with vertical structure to the velocity in the mixed layer captures these processes (Hallberg, 2004)
ASREX Mixed Layer Observations(Figures courtesy A. Gnanadesikan)
Is the bulk model’s fundamental assumption that the mixed layer is well mixed valid?
• Mixed layers do tend to be well mixed in properties such as temperature.
• Momentum is not well mixed, giving Ekman spirals with enough averaging.
The TKE budget formalism for a bulk mixed layer is probably not too bad.
A mixed layer TKE budget
Bulk mixed layer entrainment is governed by a Turbulent Kinetic Energy balance:
The mn are efficiencies and/or vertical decay of TKE. m2 is well known (1 or ~.2), while the others are not.
• Many climate models use KPP, which effectively uses TKE balance considerations to determine the mixed layer depth and diffusivity profiles. Others use TKE budgets more directly.
• Within the mixed layer, from law-of-the-wall and dimensional analysis, viscosity goes as
Adv
21
2
22
0
032
21
3*0
S
dzh
zug
meeh
BBh
muRimwumgh
wh
hhPenBulkEE
*3** /,/,/)( uzBhzfzuFzuz
1-D Mixed Layer Simulations of SST at Bermuda0.1 m Resolution KPP vs. 10 m Resolution KPP vs. GOLD with 63 layers
High-frequency specified-flux forcing, including diurnal cycle, from BATS.Equivalent initial conditions.
Sea Surface TemperatureMixing Layer Depth
Summary
Surface Planetary Boundary Layer:Several approaches seem to work well enough KPP 2-Equation turbulence closures Bulk mixed layers
Resolved-shear mixing: Many existing parameterizations in climate
models are indefensible.
Better forms may exist, but the most important open questions arise from limited climate model resolutions.