### Lecture Note

```An Eddy Parameterization
Challenge Suite: Methods for
Diagnosing Diffusivity
Scott Bachman
With Baylor Fox-Kemper
NSF OCE 0825614
Outline
• Motivation
• Math and Extant Parameterizations
• The models in the suite
Figure courtesy of Baylor Fox-Kemper
The evolution of a Parameterization
1. Figure out that you need a parameterization (Bryan 1969)!
1. Discover your parameterization isn’t doing well… (Sarmiento
1982)
1. Improve theory: physical reasoning + mathematical form (Redi,
1982; Gent and McWilliams, 1990; Gent et al., 1995; Griffies,
1998)
1. Test the theory (Danabasoglu and McWilliams, 1995; Hirst and
McDougall, 1996)
1. Validate against observations / improve theory
5. Validate against observations / improve theory
PROBLEM: What observations?
In real life, getting these observations is expensive, technically difficult,
time-consuming, and you would need a LOT of them…
To complicate matters, it is not even clear how to apply observations to
the values we use in a model (Marshall and Shuckburgh, 2006)…
But computers can help…
What if we were to “build” a suite of eddyresolving models that could tell us how a
parameterization should look? (i.e. act as
“truth” (McClean et al., 2008)
Images courtesy of Julie McClean,
SIO
An Eddy Parameterization Challenge Suite
• Primitive equation model (MITgcm)
• Lots of individual simulations
• Explore parameter space
– Vary Ro, Ri, Charney-Green, etc.
• Lots of shear/strat configurations
• Look at stirring tensor elements
– Direction of diffusion
– Dependence on parameters
– Tracer fluxes
What does this all mean?
Outline
Outline
• Motivation
• Math and Extant Parameterizations
• The models in the suite
• (Early) Results: Exponential
Relates the subgridscale eddy fluxes to coarsegrid gradient
Is the basis for most modern parameterization
methods (GM90, Redi, etc.)
GOAL: We want R.
What do we know about the
transport tensor R ?
• R varies in time and space
• R is 3 x 3 in three dimensions (2 x 2 in 2D)
• The structure of R should be deterministic
• Based on physics or the phenomenology of turbulence
• Not stochastic (…)
Lots of questions!
- Is such a closure even possible?
- Physical assumptions correct?
- Boundary layer tapering
- Rotational fluxes?
- How strong is kappa?
- Vertical structure?
And most importantly…
How do these things change in
different flow regimes?
What do we know about the
transport tensor R ?
Three things:
1) Redi (1982)
2) Gent and McWilliams (1990)
3) Dukowicz and Smith (1997) and
Griffies (1998)
What do we
1) Redi (1982)
Diffusion is not geodesic, it is isopycnal
1 0 0


SI   0 1 0

0 0 0

Small isopycnal
slopes

 1


S    0

 
 x
 z
0
1

y
z






2
2
  y  
 x     
z  z  

 x
z
y

z
What do we
2) Gent and McWilliams (1990)
QUESTION: What happens if we diffuse
density (i.e. use Redi) along an isopycnal?
Then Redi alone is inadequate… there must be
another piece to this.
What do we
2) Gent and McWilliams (1990)
Construct a parameterization that diffuses layer thickness
- Amounts to an advection by a thickness-weighted velocity
u' h '
1 
u 

 h
h
h 
*



What do we
2) Gent and McWilliams (1990)
- Properties of
u*
- Nondivergent:
  u*  0
- No flow normal to boundaries:
u*  n  0 on 

- Conserves
all domain-averaged moments of density

- Conserves all domain-averaged
tracer moments
- Skew flux (TEM):
u*    0
- Conserves tracer mean, reduces higher moments
between isopycnal surfaces
What do we
2) Gent and McWilliams (1990) – GM90
- Properties of
u*
(cont.)
- Local sink of mean potential energy

- Consistent with phenomenology of baroclinic
turbulence (mesoscale eddies)
- NOT necessarily equal to downgradient PV
diffusion along isopycnals
1 
1
u 
 h    h
h 
h
*


What do we
1) Redi (1982) – diffusion (mixing)

 1


S    0

 
 x
 z
0
1
y

z






2
2
  y  
 x     
z  z  

 x
z
y

z

1 
u 
 h
h 
*


What do we
Put Redi + GM90 into the mean tracer equation…
(in isopycnal coordinates)

  1 
1
 u 
 h      Jh   

t  h 
h


GM90
Redi
What do we
3) Dukowicz and Smith (1997) and Griffies (1998)

  1 
1
 u 
K
h




  Kh   


 

t  h 
h

Compare
with….


  1 
1
 u 
 h      Jh   

t  h 
h


GM90
Redi
What do we
3) Dukowicz and Smith (1997) and Griffies (1998)
 
The thickness diffusivity coefficient is the
same as the isopycnal diffusivity coefficient (?!)

What do we do with this information?
What do we
3) Dukowicz and Smith (1997) and Griffies (1998)
  u*  0
GM90

u*    
 u*  

    
GM90 skew flux
What do we
3) Dukowicz and Smith (1997) and Griffies (1998)
Identity
a  b  c  a  (b  c)
    

GM90 skew flux


    

What do we
3) Dukowicz and Smith (1997) and Griffies (1998)
Identity
 0

a  b   a3

a2
   

a3
0
a1
a2 b1 
 
a1 b2 
0 

b3 

 0

 3

 2
3
0
1
2  x 
 
1  y    A



0 

z
 
GM90 skew flux
What do we
3) Dukowicz and Smith (1997) and Griffies (1998)

  1 
1
 u 
 h      Jh   

t  h 
h

GM90

Redi
(in z-coordinates)
This becomes

 u    A  S 
t
What do we

3) Dukowicz and Smith (1997) and Griffies (1998)
R  A S





 
0


x
   
z

0

y
   
z

x 
   
z 
y 
    
z 
 2  2 
y
x


 2  2 

 z
z 
What do we
3) Dukowicz and Smith (1997) and Griffies (1998)
FINALLY!


 
R   0

x
2
z

0

y
2
z


0


0
 2  2 
y
x


 2  2 

 z
z 
What do we
So the theory says this….


 
R   0

x
2
z

0

y
2
z


0


0
 2  2 
y
x


 2  2 

 z
z 
But we need to know that this form is correct!

Outline
• Motivation
• Math and Extant Parameterizations
• The models in the suite
Mesoscale Characteristics
•50-100 km (ocean)
•Boundary Currents
•Eddies
•Ro = O(.01)
•Ri = O(1000)
Courtesy: K. Shafer Smith
•QG-scaling OK
The eddies extend the full depth of the water column, and are
dominated by baroclinic instability.
We are going to look at properties of baroclinic instability alone.
Why?
http://www.coas.oregonstate.edu/research/po/research/chelton/index.html
Baroclinic instabilities dominate the mesoscale.
- Barotropic instabilities smaller than a deformation radius
(too small).
- Symmetric instabilities appear at Richardson numbers <
0.95 (Nakamura, 1993).
too low
- Kelvin-Helmholtz instabilities at Ri < 0.25.
Construct models focusing only on baroclinic instability
A front to make PE available for extraction
900 x 150 x 60 grid
Large Richardson number
( anything >> 1 )
= minimal barotropic component
How do we solve for R ?
What happens if we have only one tracer?
2 Equations…
Take a zonal average, and
write the system out in full:
4 Unknowns!
Underdetermined! (not unique)
How do we solve for R ?
Use multiple tracers:
> 4 Equations…
4 Unknowns!
Overdetermined!
Moore-Penrose pseudoinverse (least-squares fit)
Tracer gradients less aligned = better LS fit!
Overdetermining the system is appropriate to reduce degrees of freedom
in the zonal average.
How do we know our solution for R is any good?
If R really is the same for every tracer, we should be
able to reconstruct the flux of a tracer that was not
involved in the pseudoinversion.
ui 'b'  Rij j b
Can we produce this with our R ?
If we are careful in initializing our tracers…
The reconstruction is excellent.
Snapshot
Good! (sinusoids)
Original fluxes
Reconstructed fluxes
Snapshot
Estimates of these buoyancy fluxes have improved substantially (error is now
< 10%)… Used to be that getting error within a factor of two was the best we
could do!
The reconstruction is excellent.
 
R1 j b
v'b'


j
Outline
• Motivation
• Math and Extant Parameterizations
• The models in the suite
The Review before the Results
We think R looks like this:


 
R   0

x
2
z

0

y
2
z
So
we are going to use a
bunch of tracers that will
tell us if we are right:


0


0
 2  2 
y
x


 2  2 

 z
z 
In 2D (zonal
average)
The Results
We run 69 simulations, looking for
 

R  2 y
z



 2  2 
y
x


 2  2 

 z
z 


And what do we get?
0
Ryy
 
Rzy
Ryz 

Rzz 
RAW OUTPUT
R
Ri

Ri

Ri
Ri
To make sense of these results, we need scalings
We are interested in finding scalings of the form
Our scalings need to be put in terms of quantities
that are present in the GCM, but where do we
begin?
We should start by considering the physics of R.
Previous Work (Fox-Kemper et al., 2008)
One could pursue naïve scalings based purely on
dimensional analysis:
So that:
We find that this is inaccurate (see below).
A Better Idea: Scale to the Process
Diffusion
A diffusive tensor can be
written in terms of
Lagrangian parcel
displacements and velocities:
The antisymmetric tensor
is responsible for the
release of mean potential
energy by baroclinic
instability, so we can scale
according to the release
rate
A Better Idea: Scale to the Process
Now choose length and time scales to substitute:

N 2H

M2
H

 v'2
t


 w'2
t





N 2H
H  2 2
2
2
2 


v'

2
2  v' M  w' N 
M
M
 

H


2
2
2
2
2
 2  v' M  w' N 

H w'

M 


H  2 2
2
2 


v'
M

w'
N
2 

M
A Better Idea: Scale to the Process
Then with Dukowicz and Smith (1997), a
dimensional base scaling for R should be:
RAW OUTPUT now becomes…
R
Ri

Ri

Ri
Ri
SCALED OUTPUT
R
Ryy

Ryz

Rzy
Rzz
SCALED OUTPUT
Our 69 simulations
(~5000 data points)
suggest scaling for R
like so:
Is this consistent with the earlier theory?
Yes! Our results suggest using GM90 + Redi
should work great, provided that we set
Does it really reproduce the fluxes correctly?
Yes!
Outline
• Motivation
• Math and Extant Parameterizations
• The models in the suite
In summary…
• We are developing a suite of idealized models for testing and
evaluating eddy parameterizations.
• We parameterize mesoscale eddy diffusivity with a linear fluxgradient relationship, governed by the eddy transport tensor R.
• We are using a new tracer-based method that solves for R in a
least-squares sense.
• Results from 69 simulations provide us with scalings for each
of the elements of R.
• The model results and the scalings are consistent with extant
parameterizations (GM90 + Redi), allowing us to describe R
completely by a prescription of κ.
Just when you thought it was safe to parameterize…
• The role of rotational fluxes gets people hot and
bothered…
• The models in this part of the research are
insufficient for analyzing the vertical structure of R
• We have prescribed a dependency on the RMS eddy
velocity, for which no parameterization currently
exists
• How do we satisfy the vertical boundary
conditions? (where Dukowicz and Smith (1997)
breaks down…)
• How does the eddy diffusivity change as the grid
resolution decreases?
• How do we parameterize the diapycnal eddy fluxes?
• Does downgradient diffusion work for potential
vorticity? (more controversy)
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