turbulent properties: - vary chaotically in time around a mean value
DESCRIPTION
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 whorls Which feed on their velocity; And little whorls have lesser whorls, - PowerPoint PPT PresentationTRANSCRIPT
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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)
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'
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- 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)