'tTT

T

Turbulent properties:- vary chaotically in time around a mean value- exhibit a wide, continuous range of scale variations- cascade energy from large to small spatial scales

“Big whorls have little whorlsWhich feed on their velocity;And little whorls have lesser whorls,And so on to viscosity.” (Richardson, ~1920)

'

'

'

''

- Use these properties of turbulent flows in the Navier Stokes equations-The only terms that have products of fluctuations are the advection terms- All other terms remain the same, e.g., tUtutUtu

0

'

0'

0'

0''

zu

wyu

vxu

uzU

WyU

VxU

U

'

''

''

'

dtUd

zwu

yvu

xuu

''''''

zw

uyv

uxu

uzu

wyu

vxu

u

'

''

''

''

''

''

'

zw

yv

xu

u'''

'

0

'','','' wuvuuu are the Reynolds stressesReynolds stresses

arise from advective (non-linear or inertial) terms

Turbulent Kinetic Energy (TKE)

An equation to describe TKE is obtained by multiplying the momentum equation for turbulent flow times the flow itself (scalar product)

Total flow = Mean plus turbulent parts = 'uU

Same for a scalar: 'tT

Turbulent Kinetic Energy (TKE) Equation

ijijoj

ijiijijij

oji eew

gxU

uueuuuupx

udtd

22

1 2212

21

Multiplying turbulent flow times ui and dropping the primes

2

21

221

221

221

wdtd

vdtd

udtd

udtd

i

Total changes of TKE Transport of TKE Shear Production

Buoyancy Production

ViscousDissipation

i

j

j

iij x

u

xu

e21

fluctuating strain rate

Transport of TKE. Has a flux divergence form and represents spatial transport of TKE. The first two terms are transport of turbulence by turbulence itself: pressure fluctuations (waves) and turbulent transport by eddies; the third term is viscous transport

zU

wu

yU

vu

xU

uu

xU

uuj

iji

wg

o

22

242

2i

j

j

i

i

j

j

iijij x

u

xu

x

u

xu

ee

interaction of Reynolds stresses with mean shear;

represents gain of TKE

represents gain or loss of TKE, depending on covarianceof density and w fluctuations

represents loss of TKE

zU

uwwg

o

0

In many ocean applications, the TKE balance is approximated as:

The largest scales of turbulent motion (energy containing scales) are set by geometry:- depth of channel- distance from boundary

The rate of energy transfer to smaller scales can be estimated from scaling:

u velocity of the eddies containing energyl is the length scale of those eddies

u2 kinetic energy of eddies

l / u turnover time

u2 / (l / u ) rate of energy transfer = u3 / l ~

At any intermediate scale l, 31l~lu

But at the smallest scales LK,

413

L Kolmogorov length scale

Typically, 356 1010 mW so that mLK

43 10610~

Shear production from bottom stressz

u

bottom

Vertical Shears (vertical gradients)

3

2

s

m

z

Uwu

Shear production from wind stressz

W

u

Vertical Shears (vertical gradients)

3

2

s

m

z

Uwu

Shear production from internal stressesz

u1

Vertical Shears (vertical gradients)

u2

Flux of momentum from regions of fast flow to regions of slow flow

3

2

s

m

z

Uwu

zU

Awu z

Parameterizations and representations of Shear Production

2

*

refB U

uC

2* refBB UCu Bottom stress:

0*

ln1

zz

uU

Near the bottom

Law of the wall

Bu *

0

* lnz

zuu

m005.0

sm04.0

0

*

z

u

Bu *

Pa2B

Data from Ponce de Leon Inlet

FloridaIntracoastal Waterway

Florida

0033.07.0

04.022

*

refB U

uC

Law of the wall may be widely applicable

(Monismith’s Lectures)

Ralph

Obtained from velocity profiles and best fitting them to the values of z0 and u*

(Monismith’s Lectures)

2

*

refB U

uC

BC

wuzz

UA

z z

wvzz

VA

z z

Shear Production from Reynolds’ stresses

Mixing of momentum

wszz

SK

z z

Mixing of property S

sm

RiK

sm

RiA

z

z

2

23

2

21

33.31

06.0

101

06.0

Munk & Anderson (1948, J. Mar. Res., 7, 276)

sm

Ri

AK

sm

RiA

zz

z

25

242

1051

1051

01.0

Pacanowski & Philander (1981, J. Phys. Oceanogr., 11, 1443)

With ADCP:

cossin4

varvar 43 uuwu

and

cossin4varvar 21 uu

wv

θ is the angle of ADCP’s transducers -- 20ºLohrmann et al. (1990, J. Oc. Atmos. Tech., 7, 19)

zV

wvzU

wuTKE Production

wuzU

Az

wvzV

Az

Souza et al. (2004, Geophys. Res. Lett., 31, L20309)

(2002)

wu

wv

Day of the year (2002)

Souza et al. (2004, Geophys. Res. Lett., 31, L20309)

Souza et al. (2004, Geophys. Res. Lett., 31, L20309)

S1, T1

S2, T2

S2 > S1

T2 > T1

Buoyancy Production fromCooling and Double Diffusion

wg

o

Layering Experiment

http://www.phys.ocean.dal.ca/programs/doubdiff/labdemos.html

wg

o

From Kelley et al. (2002, The Diffusive Regime of Double-Diffusive Convection)

Data from the Arcticw

g

o

Layers in Seno Gala

wg

o

/s)(m seawater of viscosity kinematic the is

3...1,;2

2

2

jix

u

xu

tensorratestrain

i

j

j

i

Dissipation from strain in the flow (m2/s3)

turbulence

isotropic for

5.72

zu

(Jennifer MacKinnon’s webpage)

From:

Rippeth et al. (2003, JPO, 1889)

Production of TKE

Dissipation of TKE

http://praxis.pha.jhu.edu/science/emspec.html

Example of Spectrum – Electromagnetic Spectrum

(Monismith’s Lectures)

KSS ,

Wave number K (m-1)

S (

m3

s-2)

3

2

s

m

2

3

s

mS

m

K1

3532 KS

Other ways to determine dissipation (indirectly)

Kolmogorov’s K-5/3 law

(Monismith’s Lectures)

3532 KS

P

equilibrium range

inertialdissipating range

Kolmogorov’s K-5/3 law

3532 2

U

fS

325102 sm

(Monismith’s Lectures)

Kolmogorov’s K-5/3 law -- one of the most important results of turbulence theory

Stratification kills turbulence

25.02

2

22

S

N

zv

zu

zg

Ri o

In stratified flow, buoyancy tends to:

i) inhibit range of scales in the subinertial range

ii) “kill” the turbulence

(Monismith’s Lectures)

U3

oLU 2

325101 sm

mL

zzgN

03.0,18.0,1

10/10,1,1.0 taking;

0

2

(Monismith’s Lectures)

(Monismith’s Lectures)

(Monismith’s Lectures)

(responsible for dissipation of TKE)

At intermediate scales --Inertial subrange – transfer of energy by inertial forces

nsfluctuatio of numberwave K

TKE of ndissipatio

1.5 constant

KS

3532

(Monismith’s Lectures)

3

2

sm

Other ways to determine dissipation (indirectly)