Transcript Hallberg - University of Warwick

Representing Gravity Current Entrainment
in Large-Scale Ocean Models
Robert Hallberg (NOAA/GFDL & Princeton U.)
With significant contributions from Laura Jackson, Sonya Legg,
and the Gravity Current Entrainment Climate Process Team:
NOAA/GFDL: S. Griffies, R. Hallberg, S. Legg, L. Jackson*
NCAR:
G. Danabasoglu, P. Gent, W. Large, W. Wu*
U. Miami:
E. Chassingnet, T. Ozgokmen, H. Peters, Y. Chang*
WHOI:
J. Price, J. Yang, U. Riemenschneider*
Lamont Doherty: A. Gordon
George Mason: P. Schopf
Princeton U.:
T. Ezer
(Plus ~12 active collaborators)
*Postdocs funded by the CPT
http://www.cpt-gce.org
An Idealized Rotating Overflow
DOME Test Case 1 (Legg, et al., Ocean Modelling, 2006)
Near-bottom tracer concentration
with contours of buoyancy
Tracer concentration just west of the inflow
Dx=500m, Dz=30m MITgcm Simulation
An Idealized Rotating Overflow
DOME Test Case 1 (Legg, et al., Ocean Modelling, 2006)
Near-bottom tracer concentration
with contours of buoyancy
Tracer concentration just west of the inflow
Dx=500m, Dz=30m MITgcm Simulation
Important Processes in Overflows
Resolvable by large-scale models
1. Hydraulic control at sill
2. Geostrophic adjustment of plume
along slope
3. Downslope transport of dense
water (some model types?)
4. Some geostrophic eddy effects?
5. Detrainment at neutral density
Require Parameterization
1. Exchange through subgridscale straits
2. Shear instability and entrainment
(TURBULENCE!!!)
3. Bottom boundary layer mixing and drag
processes (TURBULENCE!!!)
4. Some eddy effects?
5. Flow down narrow channels?
Shear instability &
entrainment
Geostrophic eddies
Hydraulic
control at sill
Bottom friction
Bottom-stress mixing
Downslope descent
Detrainment
Physical processes in overflows
y
z
x
Overview
•
A tour of overflows
 Oceanic Gravity Currents are important in the formation and transformation
of the majority of deep water masses.
•
Important Processes in Typical Oceanic Dense Gravity Currents:






•
Hydraulic or tidal control of source water flows, often in narrow straits
Downslope descent (gravitational, Ekman driven, and eddy induced)
Shear-driven mixing at the plume top
Bottom boundary layer mechanical stirring within the plume
Thermobaric influences of the ocean’s nonlinear equation of state
Detrainment at the neutral depth
Challenges for representing overflows in large-scale models:





Avoiding inherent problems with excessive numerical entrainment
Source water supply (representing the unresolved)
Studies of equilibrium stratified shear instability.
A new shear-driven turbulence mixing parameterization
A new bottom-turbulence mixing parameterization
Mediterranean Outflow Plume
•
•
•
Without its 3-fold entrainment, Mediterranean Outflow
water would fill the bottom of the Atlantic
Gibraltar itself exhibits rectified tidal exchange in
conjunction with hydraulic control
Because of thermobaricity, salty Mediterranean water
has a greater density at lower pressures, contributing to
shallow detrainment.
Gibraltar Velocities over the Tidal Cycle
(CANIGO cruises Send & Baschek, JGR 2001)
Climatological Salinity at 1000 m Depth
Faroe Bank Channel and Denmark Strait Outflows
Density along axis of Faroe Bank Channel
Denmark Strait Sea Surface Temperature
Denmark Strait: J. Girton; FBC: C. Mauritzen, J. Price
Abyssal Overflows – the Romanche Fracture Zone
Potential Temperature along Romanche Fracture Zone
Potential Temperature at 5000 m Depth
Ferron et al., JPO 1998
Steps in Adequately Representing Gravity Currents
1. Supply source water to the plume with the right rate and
properties.
2. Model must be able to represent downslope flow without
excessive numerical entrainment.
3. Parameterize entrainment & mixing to the right extent.
4. Parameterize subgridscale circulations? (e.g. eddies, flow in small
channels).
Shear instability &
entrainment
Geostrophic eddies
Hydraulic
control at sill
Bottom friction
Bottom-stress mixing
Downslope descent
Detrainment
Physical processes in overflows
y
z
x
Source water supply
• Source Water properties depend on the right large-scale circulation and
properties.
• Several important source waters enter through very narrow channels!
 Gibraltar is ~12 km wide.
 Red Sea outflow channel is ~5 km wide.
 Faroe Bank channel is ~15 km wide at depths that matter.
Channels that are much smaller than
the model grid require special
treatment – e.g. partial barriers.
The topography around Gibraltar, with a 1° grid (black), and the
coastline (blue) that GFDL’s 1° global isopycnal model uses.
Representing Straits with Partially Open Faces
(Work with A. Adcroft, GFDL)
Partially open faces can dramatically improve simulations of
overflows that pass through narrow straits.
The model equations need to be modified to be energetically
consistent. E.g. Sadourny’s 1975 Energy conserving
discretization of the shallow water equations:
Au  Dux Duy
Av  Dvx Dvy
Ah  Dhx Dhy
h 1
+
d iU + d jV = 0
t Ah


U = Duy uh i
V = Dvx vh j
j
i
j 
u 1
1 
1 
- u qV i = u d i  M + h  Au u 2 + Av v 2 
t D x
Dx 
2A 

v 1
+
qU
t Dvy
q=
j
i
=
1
Dvy

d j M +

j
1  u 2i
v 2 
A
u
+
A
v


h
2A 



 f + 1 d v - 1 d u
j 
i, j 
q i
q
D
D
h
x
y

A h 
Ah
i, j
Terms underlined in red are affected directly
by using the partially open faces.
Terms underlined in blue are affected
indirectly (i.e. no code changes).
Resolution requirements for avoiding numerical entrainment
in descending gravity currents.
Z-coordinate:
Require that Dz  H BBL / 2  50m
AND Dx  H BBL / 2  5km
to avoid numerical entrainment.
(Winton, et al., JPO 1998)
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?
 No existing scheme is entirely satisfactory!
Sigma-coordinate: Avoiding entrainment requires that DDOcean  H BBL 
Isopycnal-coordinate: Numerical entrainment is not an issue - BUT
• If resolution is inadequate, no entrainment can occur. Need D  12 Overflow -  Ambient
Diapycnal Mixing Equations in Isopycnic Coordinates
•
In isopycnic coordinates, diapycnal diffusion is nonlinear
h
     
z
  z 
     
=
where
h
=



  = -  

t
    h 

t   
    z 
•
The discrete form leads to a coupled set of nonlinear differential equations
hk
1   k +1D k +1  k D k 
1   k D k  k -1D k -1 

 +


=t
D k + 1  hk +1
hk  D k - 1  hk
hk -1 
2
2
These can be solved implicitly and iteratively, with an arbitrary distribution of diffusivities
to avoid the impossible time-step limit Dt / h2  1 (Hallberg, MWR 2000)
2
•
The work-diffusivity relationship is exact in density coordinates. Work =  gd = gD
•
Entrainment can also be parameterized directly, based upon resolved shear Richardson
numbers and a reinterpretation of the Ellis & Turner (1959) bulk Richardson number
parameterization (Hallberg, MWR 2000).
0.8 - Ri

Dgh
2
2
0.1DU
2
Ri  0.8
1 u -u


Ri
=
D
U
=
+
u
u
wE = 
k
k
1
k
+
1
k
1 + 5Ri
2


DU 2

0
Ri

0
.
8


This parameterization gives entraining gravity currents that are qualitatively similar to
observations, but has subsequently been improved upon.

Constant Diffusivity
Richardson Number Mixing
DOME Model Intercomparisons and Resolution
Dependence (Legg et al., Ocean Modelling 2006)
MITgcm (Z-coordinate) with Convective Adjustment
2.5 km x 60 m
10 km x 144 m
50 km x 144 m
HIM (isopycnal coordinate) with shear Ri# param.
10 km x 25 Layer
50 km x 25 Layer
Final Plume Buoyancy (m s-2)
Entrainment Rate Near Source (nondim.)
Plume Entrainment as a Function of Resolution
for 6 DOME Test Cases
Horizontal Grid Spacing (km)
Solid lines: MITgcm (Z-coordinate)
Dashed lines: HIM (Isopycnal coordinate)
For full details, see Legg et al., Ocean Modelling 2006.
Horizontal Grid Spacing (km)
Parameterizing Overflow Entrainment:
Observations of Bulk Entrainment in Oceanic
and Laboratory Gravity Currents (J. Price)
A bulk entrainment law applies, provided the Reynolds number is not quite small.
Examples of Gravity Current Mixing Parameterizations:

•
Generic shear parameterizations – e.g. KPP (Large et al., 1994):
 = 50  10 - 4 m 2 s -1 1 - Ri 0.7 2
Typically calibrated for the Equatorial Undercurrent.
Ri = N 2 / S 2
•
•
Two-equation turbulence closures (e.g. Mellor-Yamada; k-e; -w).
Plume-specific parameterizations – e.g. Ellison & Turner (1959) bulk Ri# parameterization
0.8 - Ri
reinterpreted for shear Ri# (Hallberg, 2000):

0.1DU
Ri  0.8
wEntraining = 
1 + 5Ri

0
Ri  0.8

 o DU 3  0.8 - Ri 
0.1
 Ri
This can be cast as a diffusivity, D is over an unstable region:  =
gD
 1 + 5 Ri 
May Need Resolution
Dependence!

3
Simulated Mediterranean Outflow Plume
(Papadakis et al., Ocean Modelling 2003)
Zonal Velocity
Salinity
Salinity in 3 Isopycnal Layers
A Non-rotating Overflow Entering a Stratified
Environment (Courtesy T. Özgökmen)
LES and Parameterized Overflow Entrainment
(Xu, Chang, Peters, Özgökmen, and Chassignet, Ocean Modelling in press)
Failure and Success of Existing 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 HIM-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.
2-equation (e.g. Mellor-Yamada, k-e, or k-w) closure models may be adequate.
 The TKE equations are well-understood, but the second equation (length-scale, or
dissipation rate, or vorticity) tend to be ad-hoc (but fitted to observations)
 Need to solve the vertical columns implicitly in time for:
1.
2.
3.
4.


TKE
Dissipation/vorticity
Stratification (T & S)
(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 obvious
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.
3-DNS of Shear Instability
(L. Jackson, R. Hallberg, & S. Legg in prep.)
z
3D stratified turbulence
Representative instantaneous along-channel
Cross-section in statistical steady state
Temperature (°C)
Temperature (°C)
Temperature during initial development
instability
ofKelvin-Helmholtz
Kelvin-Helmholtz instabilities
x
z
x
Considerations for a Parameterization of Stratified
Shear Instability
S = ||∂U/∂z|| [s-1]
N2 = -g/ ∂/∂z [s-2]
H [m]
Q [m2 s-2]
u* = (t/)1/2 [m s-1]
z* [m]
Velocity shear
Buoyancy Frequency
Vertical extent of small Ri
Turbulent kinetic energy per unit mass
Friction velocity (for boundary turbulence)
Distance from boundary (for boundary turbulence)
•
Mixing should vanish if the shear Richardson number (Ri = N2/S2) exceeds ~1/4
•
Vigorous mixing may extend past the region of small Ri.
•
Homogeneous stratified turbulence is often characterized by the buoyancy length scale
L2Buoy = Q / N 2
•
Kelvin-Helmholtz (K-H) saturation velocity scales are ~ H S.
everywhere
 K-H instabilities span the region of small Ri, i.e. length scales of ~ H.
 Mixing-length arguments suggest peak K-H-type diffusivities scaling as ~ H2S.
•
Near solid boundaries, length scales are proportional to the distance from the boundaries,
and diffusivities are ~ 0.4u*z*.
Translating “Entrainment Rate” parameterizations into
diffusive parameterizations (L. Jackson)
The diffusion of density can be linked to entrainment parameterizations by combining the
density conservation equation: D
   
Dt
=


z  z 
with the continuity equation in density coordinates:
ET parameterisation (Hallberg, 2000)

  z 
u
 z   1   1     

F Ri 
 -  +    - u =

  = 2
t    

    z z   z z  z  
The latter equality is ill-behaved when ∂/∂z=0, but with constant stratification it reduces to

u
= -2
F ( Ri )
2
z
z
 2
Entrainment-law derived theory for Shear-driven mixing
S=||Uz||
 2
z
2
-

L2Buoy
= -2SF ( Ri )
 = 0 at solid boundaries
LBouy = Q1/ 2 / N
Properties:
• Uses a length scale which is a combination of the width of the low Ri region (where
F(Ri)>0) and the buoyancy length scale LBuoy = Q1/2/N.
• Decays exponentially away from low Ri region
• Vertically uniform, unbounded limit:   SL2Buoy
• Ellison and Turner limit (large Q) reduces to form similar to ET parameterisation
• Unstratified limit: similar to law-of-the-wall theories of parabolic diffusivity between
two boundaries and log-like profiles of velocity near the boundaries.
TKE Budget to Complement Proposed Diffusivity Equation



Q 
 DQ 
2
2





+

+

S
N
c
Q
Q
N
=
0


=

0
0
z 
z 
Dt


Assumptions:
• Q reaches steady state faster than background flow is evolving so no DQ/Dt term
• Assume Pr = 1 (for now)
• Q0 needed to avoid singularity in diffusivity equation (solution not sensitive to Q0 and 0)
• Parameterization of dissipation as c(Q-Q0)N
LOZ =
e 1/ 2
N 3/ 2
Q1 / 2
 Lb =
N
Q intended for use in diffusivity equation is due to turbulent kinetic energy only - difficult to
compare to results from DNS because of internal waves.
Equilibrium DNS of Shear-driven Stratified Turbulence
•
•
•
•
•
•
•
•
•
Non-hydrostatic direct numerical simulations (MITgcm)
2m x 2m x 2.5m with grid size ~ 2.5mm in centre
Molecular viscosity and diffusivity, Kolmogorov scale mostly resolved.
Cyclic domain in x,y
Shear and jet profiles
Statistically steady state reached
Force average velocity profiles to evolve to given profile
Initially constant stratification and relaxed to initial density profile
All profiles are spatially averaged in x and y and time averaged
3-DNS of Shear Instability
(L. Jackson, R. Hallberg, & S. Legg in prep.)
z
3D stratified turbulence
Representative instantaneous along-channel
Cross-section in statistical steady state
Temperature (°C)
Temperature (°C)
Temperature during initial development
instability
ofKelvin-Helmholtz
Kelvin-Helmholtz instabilities
x
z
x
Buoyancy flux (m2/s3)
DNS Shear-Instability Results and the Proposed Parameterization
DNS data
F(Ri) = 0.15*(1-Ri/0.25)/(1-0.9*Ri/0.25), c=1.9
New parameterisation (Jackson et al.)
F(Ri) = 0.15*(1-Ri/0.8)/(1+1.0*Ri/0.8), c=1.7
ET parameterisation (Hallberg 2000)
F(Ri) = 0.12*(1-Ri/0.25)/(1-0.9*Ri/0.25), c=1.24
DNS of Shear Instability and Existing 2-equation closures
Black: DNS Results
Green: GOTM k-w
Blue: GOTM k-e
Red: Mellor-Yamata 2.5
Buoyancy flux (m2/s3)
Jackson et al., proposed parameterization:
Buoyancy flux (m2/s3)
DNS Jet results
DNS data
F(Ri) = 0.15*(1-Ri/0.25)/(1-0.9*Ri/0.25), c=1.9
New parameterisation (Jackson et al.)
F(Ri) = 0.15*(1-Ri/0.8)/(1+1.0*Ri/0.8), c=1.7
ET parameterisation (Hallberg 2000)
F(Ri) = 0.12*(1-Ri/0.25)/(1-0.9*Ri/0.25), c=1.24
Existing 2-equation Closures Compared to DNS Jet
Black: DNS Results
Green: GOTM -w
Blue: GOTM -e
Red: Mellor-Yamata 2.5
Buoyancy flux (m2/s3)
Jackson et al., proposed parameterizations:
Diagnosed diapycnal diffusivity (m2 s-1)
Diapycnal Diffusivities Diagnosed from 3-D DNS
Gradient Richardson number
Buoyancy flux (m2/s3)
Shear Instability with a Larger Ri#
Not yet equilibrated?
Illustrating the power
of the CPT paradigm
500 m x 30 m MITgcm
At the start of the CPT, with thick,
nonrotating plumes entering ambient
stratification, GFDL’s Isopycnal coordinate
model (HIM) would give plumes that split
in two.
Such split plumes do not occur in
nonhydrostatic “truth” simulations.
10 km x 25 layer HIM
Ellison & Turner Mixing Only
Observed profiles from Red Sea plume from RedSOX (H. Peters)
Actively mixing
Interfacial Layer
Shear Ri# Param.
Appropriate Here.
Well-mixed
Bottom Boundary
Layer
Bottom Boundary Layer Mixing
•
Diapycnal mixing of density requires work.
•
The rate at which bottom drag extracts energy from the resolved flow is
straightforward to calculate.
•
Assumptions:
 20%? of the extracted energy is available to drive mixing.
0.4u *
 Available work decays away from the bottom with e-folding scale of
f
 Mixing completely homogenizes the near bottom water until the energy
source is exhausted.
Work =  gd = gD

3
gD = ocD uBBL


 0 .4 h

BBL c D u BBL
 = 0.2 exp f


2




Legg, Hallberg, & Girton, Ocean Modelling, 2006
500 m x 30 m MITgcm
With thick plumes, both Interfacial and
and Drag-induced Mixing are needed.
(Legg et al., Ocean Modelling, 2006)
10 km x 25 layer HIM
Ellison & Turner + Drag Mixing Ellison & Turner Mixing Only
Year 5 salinity along 38.5°N in GFDL’s 1° Global Isopycnal Model
Adding the Legg et al.
bottom-drag mixing
parameterization leads to
dramatic improvements in
an IPCC-class ocean model.
Double Mediterranean
plumes without
bottom-drag mixing
Summary
•
Overflows are critical in the formation of most deep-ocean water masses.
•
Turbulent mixing with the right rate is critical for models to obtain the right properties.
(Otherwise in a stratified ambient environment, the plunging plume entrains the wrong water.)
•
Large-scale models require parameterizations of such mixing that capture both the
equilibrium turbulence and (sometimes) its equilibrium modification of the resolved flow.
•
Mature Kelvin-Helmholtz-like mixing is significant in the interfacial layers atop gravity
currents.
 Existing parameterizations do not appear to work very well in detail based on
comparisons with DNS (although they may work well enough for some overflows).
 Laura Jackson (Princeton/GFDL CPT postdoc) has a new 2-equation (diffusivity –
TKE) shear-driven turbulent mixing parameterization that looks very promising.
•
Bottom-stress driven turbulence is significant for homogenizing the bottom boundary
layer, and must be parameterized.
http://www.cpt-gce.org