Wave Modeling and Langmuir Mixing

Adrean WebbBaylor Fox-Kemper

University of Colorado

December 5, 2008

In Collaboration with: Erik Baldwin-Stevens, Greg Chini, Gokhan Danabasoglu, BenHamlington, Keith Julien, Edgar Knobloch, William Large, SyntePeacock

Research funded by: NASA NNX09AF38G & CIRES IRP08

Inverse Turbulent Langmuir Mixing Number

The inverse turbulent Langmuir mixingnumber accounts for nonaligned windand wave fields.

It is defined as

Lai =

(Ustokes · u∗|u∗|2

)1/2

, |θ| < π/2;

0, |θ| ≥ π/2.

where θ is the difference in wind and wave directions

Previous Work: A Simple Climatology

• Used output fromNWW3 to estimateareas of Langmuirmixing and derive asimple climatology

ReferencesD’Asaro, E.: 2001, Turbulent vertical kinetic energy in the ocean mixed layer. Journal of Physical Oceanography, 31, 3530–3537.

Harcourt, R. R. and E. A. D’Asaro: 2008, Large-eddy simulation of langmuir turbulence in pure wind seas. Journal of Physical Oceanography, 38, 1542–1562.

Large, W., J. McWilliams, and S. Doney: 1994, Oceanic vertical mixing: A review and a model with a vertical k-profile boundary layer parameterization. Rev. Geophys., 32, 363–403.

Li, M. and C. Garrett: 1997, Mixed layer deepening due to Langmuir circulation. Journal of Physical Oceanography, 27, 121–132.

Li, M., K. Zahariev, and C. Garrett: 1995, Role of Langmuir circulation in the deepening of the ocean surface mixed-layer. Science, 270, 1955–1957.

McWilliams, J. C., C. H. Moeng, and P. P. Sullivan: 1999, Turbulent fluxes and coherent structures in marine boundary layers: Investigations by large-eddy simulation. Air-Sea Exchange: Physics,Chemistry, Dynamics, and Statistics, G. Geernaert, ed., Springer, 507–538.

McWilliams, J. C. and P. P. Sullivan: 2001, Vertical mixing by langmuir circulations. Spill Science & Technology Bulletin, 6, 225–237.

McWilliams, J. C., P. P. Sullivan, and C.-H. Moeng: 1997, Langmuir turbulence in the ocean. Journal of Fluid Mechanics, 334, 1–30.

— 2007, Surface gravity wave effects in the oceanic boundary layer: large-eddy simulation with vortex force and stochastic breakers. Journal of Fluid Mechanics, 593, 405–452.

Pierson, Jr., W. J. and L. Moskowitz: 1964, A proposed spectral form for fully developed wind seas based on the similarity theory of S. A. Kitaigorodskii. Journal of Geophysical Research, 69,5181–5190.

Smith, J.: 1998, Evolution of Langmuir circulation during a storm. Journal of Geophysical Research-Oceans, 103, 12649–12668.

Smyth, W. D., E. D. Skyllingstad, G. B. Crawford, and H. Wijesekera: 2002, Nonlocal fluxes and stokes drift effects in the k-profile parameterization. Ocean Dynamics, 52, 104–115.

Sullivan, P. P., J. B. Edson, T. Hristov, and J. C. McWilliams: 2008, Large-eddy simulations and observations of atmospheric marine boundary layers above nonequilibrium surface waves. Journal ofthe Atmospheric Sciences, 65, 1225–1245.

Weller, R. A. and J. Price: 1988, Langmuir circulation within the oceanic mixed layer. Deep-Sea Research, 35, 711–747.

VII. ConclusionWe have demonstrated that Langmuir mixing is important in global climate models, but the resultsare sensitive to implementation details and variations in wave-wind conditions. Thus, ongoingwork will develop a reliable parameterization, improve data analysis, and incorporate WaveWatchIII as a CCSM model component.

Students SupportedAdrean Webb, CU Applied Math PhD CandidateErik Baldwin-Stevens, CU Aerospace Engineering Master’s Candidate

Papers in PreparationWebb, Baldwin-Stevens, Danabasoglu, Fox-Kemper, Hamlington, Large, Peacock: Global ModelSensitivity to Estimated Langmuir Mixing in preparation for JGR-OceansWebb, Baldwin-Stevens, Fox-Kemper: Estimating Stokes Drift from Mean Wave Variables, inpreparation for Ocean Modelling

Funded to Continue Related WorkNASA 08-PO08-0011: Langmuir Circulations: Observing and Modeling on a Global Scale($774k, $482k to CU), PIs: Fox-Kemper, Julien, Chini, KnoblochNSF OCE-0934737: CMG Collaborative Research: Multiscale Modeling of the Coupling be-tween Langmuir Turbulence and Submesoscale Variability in the Oceanic Mixed Layer($1.4M, $502k to CU), PIs: Fox-Kemper, Julien, Chini, D’Asaro & Harcourt (UW)CU ISG: Small Waves, Big Climate: Effects of Surface Gravity Waves on Climate ($21k)

AcknowledgmentsThanks to W. R. Large, S. Peacock, and G. Danabasoglu for implementing and testing the entrain-ment parameterization in CCSM 3.5 and NOAA for use of WaveWatch III output data. Thanksto Ben Hamlington for extracting TOPEX data from the archive.

Fig. 6: Mixed layer bias reduced.Fig. 5: Southern Ocean CFC bias reduced.

Initial testing of this parameterization in CCSM3.5 deepened the global mean mixed layer sub-stantially (≈ 10%), and dramatically improved the Southern Ocean shallow mixed layer bias.However, subsequent tests in CCSM4 revealed that this simple parameterization was extremelysensitive to the details of the climatology (Fig. 3), far beyond the accuracy of what may beinferred from data (Fig. 2). Thus, the importance of LMx on climate is now clear, but accu-rate modeling of these effects requires more work. Greater sophistication in parameterization iscertainly available (McWilliams and Sullivan, 2001; Smyth et al., 2002; Harcourt and D’Asaro,2008). These improvements will be readily implemented, but a prognostic wave model is requiredto model the spatio-temporally-evolving wave field. Ongoing work will refine our understandingand uncertainty estimates of Langmuir climatology and couple the WaveWatch III model as anew component of CCSM.

IV. Estimating a Climatology of Langmuir NumberOne potential reason for the mismatch of Langmuir circulation in observations is the diversecharacter of forcing. Sullivan (pers. comm.) finds in LES that Langmuir mixing is presentwhen wind and waves are misaligned. Circulation may even persist after wind has abated(Sullivan et al., 2008). Thus, we define a directional inverse turbulent Langmuir number:

La−1 =

�us · u∗

|u∗|2�1/2

, |θ| < π/2;

0, |θ| ≥ π/2.

Figure 3: Climatology of (La−1)2 (black ) with scat-tered data (red) and test alternatives

to take into account when θ, the difference in wind and wave directions, was not zero. As anexample of the spatial variability of Langmuir number, see the following figure.

Figure 4: Inverse turbulent Langmuir number squared, (La−2), (top) from NOAA Wave-Watch III model global output data (bottom)

V. Crude Parameterization Demonstrates ImportanceWith the Li and Garrett (1997) energetic scaling for the depth of LMx and an observational resultfor LC aspect ratio, we find a way to estimate the depth of LMx, H , from u∗ and us:

Fr =ω

NH≈ 0.6 ω ≈ V

1.5≈√

u∗us

1.5

This H is then used to deepen KPP mixing if Hkpp < H . The Climate model supplies u∗, andwe use the Lat climatology to infer us, and thereby close the parameterization.

I. AbstractGlobal wave and wind field data from AVISO and TOPEX altimetry, the ERA40 reanalysis,and the NOAA WaveWatch III model were used to formulate a climatology of the relationshipbetween wave and wind variables. This climatology, along with ideas from Li and Garrett (1997),were used to parameterize Langmuir mixing (LMx) in CCSM. 20th century simulations showsignificant, yet sensitive, effects from including LMx, with deeper mixed layers and improvedCFC concentrations in the Southern Ocean.

II. Langmuir Cells and Mixing

Figure 1: Cartoon of Langmuir Cells

Langmuir cells (LC) are small overturning cells(10-100m wide and 1-10km long) that form inthe near-surface ocean when wind and wavesare moving approximately in the same direc-tion. Depending on the speed of the wind andwaves, these cells can increase greatly increasethe amount of mixing in the mixed layer. Obser-vations indicate that even when these cells are notobvious, Langmuir turbulence (LT)–a disorderedjumble of LC–can lead to near-surface turbulent

kinetic energy double what is expected without LMx (D’Asaro, 2001). The turbulent Langmuirnumber, La = (u∗/us)

1/2 (McWilliams et al., 1997), is a non-dimensional parameter useful ininferring the additional Langmuir mixing, where u∗ is the skin friction velocity from wind andus is the Stokes drift veloctiy of the waves.

III. Potential Importance of Langmuir TurbulenceThe ocean surface acts as a filter on ocean-atmosphere communication of momentum, energy, andchemical tracers (e.g., C02) and contains the euphotic region where phytoplankton grow. Subme-soscale and smaller (¡10 km) physics create and preserve this environment, so it is important toaccurately model and parameterize these unresolved scales in this turbulent region. Before thiswork, it was unclear how important a role LMx may play in deepening the mixed layer since thisregion that is already well-mixed. Observations differ as to the importance on LMx: some showrapid deepening of the mixed layer in the presence of LC (Smith, 1998; Li et al., 1995; D’Asaro,2001), others do not (Weller and Price, 1988). Large Eddy Simulations (LES) indicate potent ef-fects of LT (McWilliams et al., 1997; McWilliams et al., 1999; McWilliams and Sullivan, 2001;McWilliams et al., 2007; Harcourt and D’Asaro, 2008).

This work set out to determine whether on a global scale conditions are sufficiently favorable forLT that they may play a frequent enough role to affect the climatology of the ocean surface layer.It was suspected that some observations might fall where these circulations were expected to beweak (Lat � 1), while others where they were strong (Lat � 1).

Currently, the NCAR CCSM model uses the KPP mixing scheme (Large et al., 1994) to accountfor near-surface mixing. LMx is included only indirectly through tuning–the parameterization istrained against data but does not include explicit wave information. It was generally supposedthe ocean wave field is usually fully-developed, so wave information could be inferred (Pierson,Jr. and Moskowitz, 1964). Fully-developed waves have Lat ≈ 1/

√10. In additon to measuring

surface height, altimeters are able to measure wave height, wind speed, and wave period. TOPEXaltimetry and ERA40 reanalysis (assimiliating ERS altimetry and wave buoy data) both indicatethat the variability of Langmuir number is likely to be much larger (see figure below).

Figure 2: Langmuir number from satellite altimetry (left), and WaveWatch III data (right).

Edgar KnoblochUC-BerkelyDept. of Physics

Keith JulienDept. of Applied Mathkeith.julien@colorado.edu

Greg ChiniU. New HampshireDept. of Mech. Eng.

Baylor Fox-KemperCIRES, ATOCbfk@colorado.edu

Erik Baldwin-StevensStudent: Dept. of Aerospace Eng.erik.baldwinstevens@colorado.edu

Adrean WebbStudent: Dept. of Applied Mathadrean.webb@colorado.edu

Windrows in Global Models: Does Langmuir Mixing Matter for Climate?

A Simple Scaling for Langmuir Depth/Entrainment:

(Li & Garrett, 1997)

Use Fr to determine H

Large came up with clever choices for N, H that lead to a robust implementation in KPP

If H is deeper than KPP Boundary Layer depth, use H

With these choices, H and BLD converge over time.

CAM

related to CAM u* by

WW3Climatology

The Algorithm

Previous Work: Shown Sensitivity to Inclusion

(a) CFC in CCSM 3.5 & P14S WOCE obs (b) August mixed layer depths

Problem 1: Calculating the Surface Friction Velocity

• Installed WW3 on bluefire (details later)

• Obtained similar calculations of Lai using WW3’s u∗ withCOREv2 forcings

Problem 2: Estimating Stokes Drift

For monochromatic waves, it can be shown that at the surface

Ustokes =π3Hs2

gTm3

where Hs = 4√

m0 and m0 is the zeroth moment of the variance.

However, this is not true for anything other than monochromaticwaves.

Problem 3: Different Definitions of Mean Wave Period

WaveWatch: Tm0 = (f −1)

ERA40: Tm1 = 1/

(f )

TOPEX: Tm2 = 1/√

(f 2)

mn =

∫ 2π

0

∫ ∞

0f n S(f , θ) df dθ

Tm0 =m−1

m0, Tm1 =

m0

m1, Tm2 =

(m0

m2

)1/2

A Quick Example

Pierson-Moskowitz Spectrum

S(f , θ) = S(f ) =αg2

(2π)4f −5 Exp

[−5

4

(fpf

)4]

where α is the Phillips constant and fp the peak frequency

(Tm0/Tm1)3 = 1.37, (Tm0/Tm2)3 = 1.76

(Tm1/Tm2)3 = 1.28

Calculating Stokes Drift Using 2-D Spectrum

From previous work by Kenyon (1969) and McWilliams & Restrepo(1999), we can reformulate Stokes drift using the 2-D spectrum as

Ustokes =16π3

g

∫ 2π

0

∫ ∞

0f 3 S(f , θ) df dθ ed

=16π3

gm3 ed

where ed is the dominant direction of wave propagation.

As a result, we no longer need the previous Ustokes approximation!

Refining our Stokes Drift Approximation

• Would still like to be able to estimate Stokes drift usingsatellite and buoy data for comparison

• Currently examining if there is an empirical or mathematicalrelationship that we can use such as

Ustokes ≈ a(f )π3Hs2

gTm3ed

Current Estimate of La2i

Problem 4: Numerical Cost

• 3rd generation wave model

• Solves the spectral action densitybalance equation

• 15-20 sec per time step (1 hr) forone processor (≈ 35-50 hr/yr)

• Plan on scaling back the number ofbins significantly and turning offsome interactions

• Aternative 2nd generation modeldeveloped by George Mellor(Princeton) worth exploring

Applications of Coupling a Wave Model

• Calculate Langmuir Mixing forcing prognostically

• A coupled wave model will allow use of more sophisticated andvalidated parameterizations (e.g., Smyth et al, 04; Harcourt &D’Asaro, 08; Grant & Belcher, 09)

• Improve the air-sea momentum flux

• Improve the air-sea tracer flux

• Conduct climate change studies like erosion

• Others?

Some Properties of a(f )