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