The structure of turbulence in a shallow water wind-driven shear current with Langmuir circulation

Andrés E. Tejada-Martínez and Chester E. Grosch

Center for Coastal Physical Oceanography

Department of Earth, Ocean and Atmospheric Sciences

Old Dominion University

Norfolk, Virginia

Observed structure of Langmuir cells

1x

2x

3x

Courtesy of J. Smith, UCSD

Negatively buoyant algae aligned in rows by Langmuir circulation off the coast of the Bahamas (courtesy of D. Zimmerman, ODU)

The filtered Navier-Stokes equations

• Continuity:

• Momentum:

2Re

HU

ksjijk

Tj

drij

i

i

ij

ij

i uLaxx

u

xx

uu

t

u

2

)(

2

2 1

Re

11

0

i

i

xu

)(

3

1 rkk

ST U

ULa

)(sinh2

)](2cosh[23

1 kH

Hxku s

032 ss uu

SGS stress

Craik-Leibovich vortex forcing

ijTijrkk

rij

drij Sv2

3

1 )()()(

jijirij uuuu )(

i

j

j

iij x

u

xu

S21

Subgrid-scale (SGS) stress

SGS stress:

Smagorinsky model for the SGS stress:

Eddy viscosity: ||)( 2 SCST ijijSSS 2||

2)( SC is computed dynamically using the Germano identity (Germano et al., Phys. Fluids, 1991)

LES of Langmuir cells in wind-driven channel

hx 3

hx 3surface

• Surface stress is applied such that 180/Re hu

• Craik-Leibovich vortex forcing is added to the filtered momentum equations (LES equations) to account for Langmuir cells (Lc)

• Two simulations were performed: 1) Langmuir forcing, 2) no Langmuir forcing

no-slip wall

hL 41 hL )3/8(2

• Parameters in simulation with Lc are derived from wave and wind conditions during field observations of Lc:

1x2x3x

,7.0TLa H6

hH 2

LES of Langmuir cells (mean streamwise velocity)

• Enhanced mixing due to Langmuir circulation tends to homogenize mean streamwise velocity

LES of LC (instantaneous streamwise velocity fluctuation)

No Langmuir forcing

With Langmuir forcing

iii uuu 1x

2x

LES of Langmuir cells (streamwise-time averages)

No Langmuir forcing

spanwise vel.fluctuations

wall-normal vel.fluctuations

streamwise vel. fluctuations

With Langmuir forcing

1x2x

3x

LES of Langmuir cells (Reynolds stress components)

No Langmur forcing

With Langmuir forcing

LES of Langmuir cells (Lumley’s triangle)

No Langmuir forcing With Langmuir forcing

• Presence of Langmuir cells greatly affects the state of the turbulence near the bottom and near the surface

• Trajectory of Lumley map for case with Langmuir forcing agrees well with observed data especially near the bottom

• See tomorrow’s talk at 9:40am: “Characteristics of Langmuir turbulence observed in shallow water,” Judith Wells and Ann Gargett

Turbulent kinetic energy budget terms (near bottom)

No Langmuir forcing

With Langmuir forcing

No Langmuir forcing

With Langmuir forcing

Turbulent kinetic energy budget terms (near surface)

Final Remarks

• First observations and simulations of Langmuir circulation covering entire water column

• Secondary flow structures of simulations characteristic of Langmuir circulation agree well with observations

• More recent work explores effects of: 1) Reynolds number 2) domain size 3) subgrid-scale parameterization 4) grid size 5) rotation 6) stratification

• Reynolds stresses of simulations agree well with observations in lower region of the water column

Domain length dependence

Extended domain

spanwise vel.fluctuations

wall-normal vel.fluctuations

streamwise vel. fluctuations

Original domain

1x2x

3x

Reynolds number dependence

Re = 180

Re = 395

SGS model (parameterization) dependence

DynamicSmagorinsky

Dynamic mixed

Grid dependence

32x64x97

48x96x145