1. Parameterizing Vertical Mixing in the Coastal Ocean
Scott Durski, Scott Glenn, Dale Haidvogel, Hernan Arango
Institute of Marine and Coastal Sciences
Rutgers University
New Brunswick, NJ
January 27, 1999
Vertical mixing parameterizations for a regional ocean circulation model
In the NOPP project where the goal is highly accurate simulations of the coastal
ocean many details of the numerical model formulation can play a critical role
in determining the success of the prediction. Early on in our development of
realistic models for the NYB we realized that our representation of vertical
mixing was one such detail.
So what I am going to talk about today is a step back - or maybe several steps
back from the realistic modeling that Hernan will talk about next. I will discuss
a simple two dimensional upwelling setting which I used as a tool for studying
vertical mixing parameterizations.
I will discuss major modifications to one popular oceanic vertical mixing
scheme for this coastal ocean setting and discuss it’s performance in comparison to another.
In this talk I will discuss model-model comparisons and suggest where
discrepancies between the two models suggest it would be fruitful to concentrate
observational efforts for further model improvements.

2. Vertical Mixing Processes Model Parameterizationsu* Kv u
ku*z
2nd and higher order closure schemes - Mellor-Yamada, Gaspar, Stull.
Boundary layer mixing
K profile parameterizations (1st order) - Troen and Mahrt, Large, McWilliams and Doney.
~ Mixed layer models - Price et al. ,Kraus and Turner.
Langmuir circulation
Surface wave breaking
Interior shear driven processes
Gradient Richardson number based schemes - Pacanowski-Philander, Large et. al(interior scheme), Kantha and Clayson.
There are numerous vertical mixing processes that can potentially play a role in
the coastal ocean. They’ve received varying degrees of attention in terms of
model parameterization.
The two typically considered most critical are boundary layer mixing processes
and interior shear driven processes.
Boundary layer mixing has rightfully received the most attention. A wealth of
laboratory and atmospheric observations have been studied. We know
approaching solid boundaries to expect log layer behavior, Monin-Obukov
similarity theory and a turbulent viscosity coefficient K behaving linearly,
(neutral case)
But in the boundary layer other processes may also be important
Surface waves and Langmuir circulation are two phenomena whose impact on
mixing have not been fully assesses.
In the oceans interior away from boundary layers we talk about shear instability
mixing, internal wave mixing … interrelated phenomena…
Discuss the different parameterizations that account for these different phenomena.
Mention how the Mellor Yamada is basically the industry standard for coastal studies.
A popular model for open ocean application tha has shown performance some greater success than the MY at matching some
data sets is the LMD KPP
...
2nd and higher order closure schemes - Mellor-Yamada, Gaspar, Stull.
Background mixing constant - Large et al.
Internal wave breaking
Proportional to N-1 - Gargett and Holloway..
Double diffusive processes
Density ratio base schemes - Large et al.

3. 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
Boundary layer depth Ko
0.5
Rig
Based on bulk Richardson number.
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’.

4. Addition of a bottom boundary layer parameterization
Modifications to the K-profile Parameterization for coastal ocean application.
Addition of a bottom boundary layer parameterization ?
1) Matching with log layer similarity theory where surface boundary layer extends to the bottom..
2) Apply matching rules when surface and bottom boundary layers overlap.
3) Add a K-profile parameterization for the bottom boundary layer modeled after SBL approximation.
Kv = ku*z
Change in internal wave mixing parameterization for interior
Replace constant value with the buoyancy frequency dependent formulation of Gargett and Holloway
Kvi = 1.0x10e-7 N-1

5. 2-Dimensional Model Setup
Circulation model
S-coordinate Rutgers University Model (SCRUM).
Idealized 2-dimensional domain
21.5
75 km horizontal extent.
depth ranges from 6m at the coast to 40m offshore.
Grid resolution varies from 500m at the coast to 4km offshore.
50 sigma coordinate vertical levels.
open boundary conditions applied at offshore boundary.
20.6
19.6
Initialization
18.7
horizontally uniform stratification.
stratification ranging from kg/m3 to kg/m3 .
17.8
Forcing
4 days of uniform 0.28 dyne along-shore upwelling favorable wind stress.

6. Across-shore velocity
Basic Model Response
Advectively dominated upwelling process.
Formation of an alongshore jet reaching a magnitude of 50 cm/s
Development of surface and bottom boundary layers
Vertical mixing ‘overcomes’ upwelling in the nearshore region before sub-pycnocline water reaches the coast
Across-shore velocity
Density
Along-shore velocity

7. Across-shore velocity
Model-Model Comparison
The basic response with the two schemes is quite similar but ....
More intermediate density water is trapped at the coast with the Mellor-Yamada scheme.
The surface jet is approximately 10 cm/s weaker with the Mellor-Yamada scheme.
Density
Across-shore velocity
Along-shore velocity
LMD
M-Y

8. The development of the upwelling front
1) The Mellor-Yamada parameterization entrains more water into the bottom boundary layer as the pycnocline advects shoreward causing the isopycnals of the bottom front to broaden
2)As a consequence of this weaker stratification occurs sooner in the near shore region with the Mellor - Yamada scheme.
3) Vertical mixing breaks down this weaker stratification forming a surface-to--bottom boundary layer earlier. This prevents further upwelling at the coast.
Diffusivity
LMD
M-Y

9. compresses isopycnals
What causes the Mellor-Yamada scheme to entrain more?
Diffusivity profiles
1) The gradient in diffusivity coefficient at the edge of the pycnocline tends to be greater with the LMD parameterization.
LMD
M-Y 
spreads isopycnals
compresses isopycnals
LMD
M-Y
smaller
larger 
2) The higher stratification formed by the compression of isopycnals, creates an increasingly intense barrier to vertical mixing
3) Mixing at the base of the boundary layer in open ocean settings is likely to be characterized by significantly weaker gradients in diffusivity and density.

10. Interior shear mixing formulations
An alternate formulation for shear generated mixing
1) The Pacanowski and Philander parameterization estimates a more gradual variation of diffusivity with gradient Richardson number between Ri=0.15 and 0.6.
diffusivity
Gradient Richardson number
LMD PP
Interior shear mixing formulations
LMD PP Ko
(1 + 5Ri)2
K =

11. Original LMD LMD w/PP interior
Comparison of original LMD shear mixing scheme with modified P-P interior.
Using the Pacanowski and Philander shear mixing term for the interior produces frontal intensity and near shore densty structure more similar to the Mellor - Yamada scheme.
Original LMD
LMD w/PP interior

12. The development of the upwelling front
Diffusivity
LMD w/PP interior
Original LMD

13. Which aspect of PP makes the difference?

14. Summary
The modifications to the Large, McWilliams and Doney scheme produce a parameterization with behavior quite similar to the Mellor -Yamada scheme for a wind driven continental shelf upwelling simulation.
Greater entrainment into the bottom boundary layer produced by the Mellor -Yamada scheme leads to the formation of a less intense upwelling front and the trapping of more intermeadiate density water at the coast.
The greater entrainment by the Mellor-Yamada scheme is due to a weaker gradient in the diffusivity coefficient where the boundary layer meets the pycnocline.
More observations of mixing at the entrainment depth will help direct future model improvement.