modelling turbulent stirring of a large stratified lake
DESCRIPTION
MODELLING TURBULENT STIRRING OF A LARGE STRATIFIED LAKE. Peter A Davies (University of Dundee, UK) William Rizk, Alan Cuthbertson (University of Dundee) Yarko Nino (University of Chile). Geophysical/Environmental context. Wind-induced hydrodynamics of stratified lakes, reservoirs - PowerPoint PPT PresentationTRANSCRIPT
MODELLING TURBULENT STIRRING OF A LARGE STRATIFIED LAKE
Peter A Davies(University of Dundee, UK)
William Rizk, Alan Cuthbertson (University of Dundee) Yarko Nino (University of Chile)
Geophysical/Environmental context
• Wind-induced hydrodynamics of stratified lakes, reservoirs
• (Csanady, 1968, 1972; Spigel & Imberger, 1980; Imberger & Hamblin, 1982; Imberger & Patterson, 1990 etc etc)
• Coastal hydrodynamics (e.g Baltic Sea)• (Walin, 1972)
Field data: Case I Lake Villarrica, Chile
• Strong, down-valley, warm, föhn-type summer winds (Puelche)
• Summer stratification (Meruane, Nino & Garreaud, 2008)
Field data – Lake Villarrica
• Puelche events (3-4 days) –thermocline distortion
Field data: Case II – Lake Kinneret
• Periodic forcing• Daily, summer sea
breeze (15 m.s-1 at 10 m)
• Internal Kelvin, Poincaré waves
Antenucci & Imberger, Limnol. Oceanogr. (2003)
(Antenucci & Imberger, 2005)
Previous laboratory modelling studiesNon-rotating cases: configuration 1
• Surface forcing: entrainment from below• Downward migration of boundary between unmixed and
mixed fluid
ρ1
ρ2
x = 0 U
∂
x = L
g ↓
ue ↓
Kranenburg, 1985; Nino et al, 2003
Previous laboratory modelling studiesNon-rotating cases: configuration 2
• Base forcing: entrainment from above• Upward migration of boundary between mixed and
unmixed fluid
U
∂
ρ1
ρ2
x = 0 x = L
Monismith (1986)
g ↓
ue ↑
Non-rotating cases: parameterisation
• Define: Ri* = g'h1,2/u*2
• u* = (τ0/ρ1,2)1/2 g'= g(ρ2 - ρ1)/ρ1 τ0 = [(μ∂‹u›/∂z) – (ρ1,2‹u´w´›)]
• Entrainment Parameterisation: ue/u* = k Ri*-n
ρ1
ρ2
x = 0
U
∂
x = L
U
∂
ρ1
ρ2
x = 0
x = L
Present model:Effects of background rotation
• Rotating container• Rigid lid, moving bottom boundary (“Configuration 2”)
L
Hh1, ρ1
h2, ρ2 U
x
y
z
Ω
Width W
Dimensionless Parameters (rotating flow)
• Ri* = g’h2/u*2; Ke-1 = Rod/W (U/u* ~ 17-20)
• [Rod = c/2Ω); c2 = g´h1h2/(h1 + h2)]
• Re = Uh2/ν (> 3.5 x 104): h1/h2 ( = 2): H/L: H/W
• Derived parameters: Ro-1 = 2ΩW/U; WN = Ri*(h2/L); Ek = ν/2Ωh22
L
Hh1, ρ1
h2, ρ2 U
Ω
Lake Villarrica: c ~ 0.54 m.s-1; Rod ~ 7.5 km; Ke-1 ~ 0.3; Ro-1 ~ 10-1
Experimental facility
U
2-layers, immiscible (saline, fresh water)
Density and velocity profiles
1 2 3
U
0.1Lx
5.0Lx
0Lx
4
5.0Wy5.0
Wy 0
Wy
5.0Lx
23.0Lx
Centre
37.0Wy37.0
Wy 0
Wy
Non-rotating casesDensity profiles – time series
Ri* = 37.5, Re = 3.4 x 104 (WN = 2.5)
3
2
1
4
U
-0.2 0 0.2 0.4 0.6 0.8 1 1.20
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
/(2-1)
z/H
Discrete density time series graph for probe 2 for Exp. (i) 17(c/L)t = 0(c/L)t = 52.6(c/L)t = 107(c/L)t = 162(c/L)t = 216
h2/h1 = 1/2
Time scale? Ω-1, L/c, L/U
Density profile time series (non-rotating cases)
Δρ/(Δρ)0 versus ct/L
•
Ri* = 16.6 , Re = 3.9 x 104, (WN = 1.1)
3
2
1
4
U
1
2
3
4
Non-rotating cases Track bounding isopycnal (Δρ)/(Δρ)0 = 0.05
•
0 50 100 150 2000
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
(c/ L)t
z/H
Ri* = 37.5, Re = 3.4 x 104
1, 2, 3, 43
2
1
4
U
Non-rotating cases: Entrainment velocity parameterisation
Note that WN = (Ri*)(h2/L)
1.0E-05
1.0E-04
1.0E-03
1.0E-02
1.0E-01
0.1 1 10 100WN
u e/u
*
Rizk et al. (2008), L/H = 5
Nino et al. (2003), L/H = 6
Nino et al. (2003), L/H = 9
Kranenburg (1985), L/H = 26
Kranenburg (1985), L/H = 74
Rotating cases: Δρ/(Δρ)0 versus ct/L
Ri* = 52.2, WN = 3.5, Ro-1 = 0.43, Ke-1 = 0.68 3
2
1
4
U
1
2
3
4
Rotating cases Δρ/(Δρ)0 versus ct/L
Ri* = 68.7, (WN = 4.6), Ro-1 = 0.50, Ke-1 = 0.68 3
2
1
4
U
1
2
3
4
Rotating casesΔρ/(Δρ)0 versus ct/L at z/H = 0.11
Ri* = 52.2 (WN = 3.5), Ro-1 = 0.43 (Ke-1 = 0.68) 3
2
1
4
U
0 100 200 300 400 500 600 7000
0.2
0.4
0.6
0.8
1
(c/ L)t
/
(2-
1)
1, 2, 3, 4
Rotating casesTrack Δρ/(Δρ)0 = 0.05 isopycnal
Ri* = 68.7, (WN = 4.6), Ro-1 = 0.50, Ke-1 = 0.68
3
2
1
4
U
0 100 200 300 400 500 6000
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
(c/ L)t
z/H
1, 2, 3, 4
Bounding isopycnal (Δρ/(Δρ)0 = 0.05)
• Longitudinal and transverse slopes• Both slopes = 0 for non-rotating cases• Non-zero slopes with rotation.
1
2
U
1
2
O U
z↑
y = Wy = 0x = L x = 0
Rotating casesSlope of Δρ/(Δρ)0 = 0.05 isopycnal
Ri* = 52.2 (WN = 3.5), Ro-1 = 0.66 (Ke-1 = 0.45)
3
2
1
4
U
0 200 400 600
-0.1
0
0.1
0.2
0.3
(c/ L)t
z/
z/ (0.35L)z/ (0.35W3)
z/ (0.35W1)
Rotating Cases: Plan view (velocity/vorticity)
z/H = 0.24; ct/L = 47.1; 0.84 < x/L < 0.16
x/L
y/W
-0.1
-0.05
0
0.05
0.1
Ri* = 21.9 (WN = 1.47); Ro-1 = 0.50 (Ke-1 = 0.33)
← U
s-1
Rotating Cases: Velocity profiles u(z)
Ri* = 21.9 (WN = 1.47); Ro-1 = 0.50 (Ke-1 = 0.33); ct/L = 47.1
0.00
0.20
0.40
0.60
0.80
1.00
1.20
-5.00 -3.00 -1.00 1.00
u/ u*
z/H
(x/ L) = 0.56
(x/ L) = 0.49
(x/ L) = 0.42
(x/ L) = 0.35
(x/ L) = 0.28
(x/ L) = 0.22
0.00
0.20
0.40
0.60
0.80
1.00
1.20
-5.00 -3.00 -1.00 1.00 3.00 5.00
u/ u*
z/H
(x/ L) = 0.56
(x/ L) = 0.49
(x/ L) = 0.42
(x/ L) = 0.35
(x/ L) = 0.28
(x/ L) = 0.22
0.00
0.20
0.40
0.60
0.80
1.00
1.20
-1.00 1.00 3.00 5.00
u/ u*
z/H
(x/ L) = 0.56
(x/ L) = 0.49
(x/ L) = 0.42
(x/ L) = 0.35
(x/ L) = 0.28
(x/ L) = 0.22
y/W = 0
y/W = -0.38
y/W = 0.38
Conditions of Geostrophy
• 2Ωu = -(1/ρ0)(∂p/∂y) → (∆z)/(∆y) ~ 2Ωu/g′
• Measurements?
• umax ~ 1-3 (u*) ~ (1-3)(U/20) ; Ω ~ 0.24 s-1; g'= 0.03 – 0.1 m.s-2
• 2Ωu/g′ ~ 0.2 - 0.5
• Note that Ro' = umax/2ΩW << 1
Rotating cases: Entrainment velocity
0.0000.0010.0020.0030.0040.0050.0060.0070.0080.0090.010
0 20 40 60 80 100 120 140
Ri*
u e/u
*
Ro(̂ -1) < 0.49
0.50 < Ro(̂ -1) < 0.59
0.60 < Ro(̂ -1) < 0.69
0.70 < Ro(̂ -1)
Probe 1
3
2
1
4
U
Rotating cases: Entrainment velocity
0.0000.0010.0020.0030.0040.0050.0060.0070.0080.0090.010
0 20 40 60 80 100 120 140
Ri*
u e/u
*
Ro(̂ -1) < 0.49
0.50 < Ro(̂ -1) < 0.59
0.60< Ro(̂ -1) < 0.69
0.70 < Ro(̂ -1)
Probe 23
2
1
4
U
Conclusions
• Strong background rotation destroys 2d response of non-rotating counterpart flows
• In rotating cases, significant transverse & longitudinal slopes of bounding isopycnal between unmixed and mixed fluid layers, with formation of boundary currents.
• Boundary currents in geostrophic balance (at least in early stages of flow development).
• Enhanced entrainment in boundary current region (lower gradient Ri?)
• Entrainment still parameterised well by Ri* in strongly rotating system
BJØRN GJEVIK – TEACHER AND ATHLETE
ca 1979