Turbulence and mixing in estuaries

Rocky Geyer, WHOI Acknowlegments:

David Ralston, WHOI Malcolm Scully, Old Dominion U.

Wind-driventurbulence

Interfacial, shear-driven turbulence

Boundary-layerturbulence

wind

velocity

Velocity =log layer

stress

Bottom stress

τB /ρ= Cdub2=u*

2Turbulent velocity scale

uT ~u* ~ 0.05 ub

ub

“eddy viscosity”

Simplest case: unstratified tidal flow:Only boundary-layer turbulence

Mixing Length model for the Eddy Viscosity / Diffusivity

2/1.0)/1(

)/1(

max

*

*

hzathhzz

uuu

hzzuK

T

T

m

Tu

from log layer observations:

define:

)(z

z

Velocity

stress

ub

“eddy viscosity”

Now add stratification

log layernear bottom

stratification damps turbulencenear pycnocline

enhanced shearnear pycnocline

ρ1

ρ2

zgN

gg

2

'“reduced” gravity

Buoyancy frequency

What is the maximum vertical scale for turbulence with stratification?

z

pgzuB 2

21

Bernoulli Function (energy-conserving flow)

Nu

uutake

zNu

zz

g

dzzgzgu

To

T

i

zh

hiii

1

2

2

)(21

)(21 1

1

The Ozmidov scale:maximum size of eddies beforegravity arrests them.

oz

i

Boundary layer: LT ~ kz

turbulence suppressed

Ozmidov scaling:LT=uT/N

Schematic of turbulence length-scale in a stratified estuary

dist

ance

from

bed

u(z)

h

0

z

LBL

Limiting Length-scales in Turbulent Flows

Boundary-Layer Scaling (depth limiting)

LT

Note that LT Thorpe overturn scale

Ozmidov Scaling (stratification limiting)

0.2 h

2/1

1

hzzLL BLT

2/1

3

N

LL OT

Relative flow direction

kosp

ectra

l den

sity

S(k

)

Turbulence length-scale LT~ 1/ko

Ozm

idov

Len

gth

scal

ing

Boundary-Layer scaling

Snohomish River

Scully et al. (2010) Influence of stratification on estuarine turbulence

Turbulent dissipation (conversion of turbulent motions to heat)

2

j

i

xu

In a boundary layer, dissipation ~ production

zuu

2*

ensembleaverage of turbulent motions

=

Dissipation: the currency ($ or € ?) of turbulence

ko

3/53/2)( kakS o

“Inertial subrange” method for estimating dissipation:

Viscous limit

10 cml o

=1 m

Turb

ulen

t Dis

sipa

tion

ε, m

2 s-3

Buoyancy Frequency N, s-1

ContinentalShelf

Ocean

Rivers

Lakes

3 cm30

cm3 m

Estuaries

The Parameter Space of Estuarine Turbulence

lo = ( ε/N3 )1/2

Geyer et al. 2008: Quantifying vertical mixing in estuaries

Stratified boundary layer

u(z)u(z)

Stratified shear layer

no turbulence

Two different paradigms of estuarine mixing. How importantis the stratified shear layer paradigm in estuarine turbulence?

turbulence

turbulence turbulence

Shear Instability

25.0

Ri 2

2

RizuN

necessary condition for stabilityMiles, 1961; Howard, 1961

gradient Richardson numberRichardson, 1920

Thorpe, 1973

Smyth et al., 2001

ub

hi

us ρ1

ρ2

xhgwu

xhg

zwu

i

i

81''

'''

max

2

2

Momentum balance of a tilted interface

0.5-1x10-4 m2s-2 for strong transition zones –moderate but not intense stress

meters 1.2 m/s

weak motion

bottom

interface

Fraser River salt wedge—early ebb (Geyer and Farmer 1989)

1.2 m/s

400 200 m 0 200 180 m 160

Ri<0.25leadingto shearinstability

Connecticut River: Geyer et al. 2010: Shear Instability at high Reynolds number

Salinity

dissipation of TKE

dissipation of salinity variance

Day 325--Transect 17 (~ hour 19.1)

riverocean

meters along river

MB

MB

MBC C C

MMM

M

B

M

B

M

BC C C

#4

#5

#6MMM

M BM BM BC C C

#4#5#6

MMM

Echo Soundingat Anchor Station

Salinity contours (black)Salinity variance (dots)

B: braidC: coreM: mixing zone

Salinitytimeseries~ 60 seconds

Staquet, 1995

Re~1,000

Re~500,000

MIXINGin cores

MIXINGin braids

Baroclinicity of the braid accelerates the shear… with plenty of time within the braid…

α

ρc

ρ2

ρ1

323max

max

11

105~8

21~8

''

7.0~Ri81

smug

Pagwu

TTg

zu

t oadv

acc

…leading to mixing:

20 seconds30 meters

New profiler data and acoustic imagery

Very intense, and very pretty…

…but is mixing at hydraulic transitions important at the scale of the estuary?

100 m

fresh

salt

NetTidal Power “P”

Buoyancy flux B = ∫∫∫β g s′w′ dV

Dissipation D = ∫∫∫ε dV

Energy balance: P = B + D

Efficiency Rf = B/P = B/(D+B)

Hudson: ROMS

Merrimack: FVCOM

Massachusetts

Boundary layer

Internal shear

U(z)U(z) u’w’u’w’

In the estuaryMerrimack River mixing analysis

volume-integrated buoyancy flux

Boundary layer

Ralston et al., 2010Turbulent mixing in astrongly forced salt wedge estuary.

Hudson River mixing analysisROMS, Qr = 300 m3/s

Scully, unpublished

Boundary layer

Boundary layer

Internal shear

Internal shear

Ri

Rf

testing turbulence closure stability functions with Mast data

Kantha and Clayson 1994

Canuto et al., 2001 Scully, unpublished

k- Mellor-Yamada 2.5 (k-kl)

ebb depth

dtdzB

Observed buoyancy flux vs. Ri

Modeled buoyancy flux vs. Ri

Conclusions and Prospects for the Future

1. Stratified boundary-layer turbulence is the most important mixing regime in estuaries.

2. Shear instability is locally important and dramatic but is not the dominant contributor to the total mixing.

3. Closure models are on the right track. We need more data for testing them.

4. Estuaries are outstanding natural laboratories for the investigation ofstratified mixing processes. We need more measurements of turbulencein these environments!