Application of modal analysis to strongly stratified lakes

1
Application of modal analysis to strongly
stratified lakes
Supervisor Jörg Imberger

• PhD candidate Kenji Shimizu

2
Overview
Force
Currents
Energy flux
Turbulence
3
Objective Contents
Objective better understanding basin-scale
motions Methodology modal analysis

• Modes? Modal analysis? Motivation
• Horizontal structure excitation of internal
  waves gyres in Lake Biwa Numerical
  simulation
• Energetics damping of internal waves in Lake
  Kinneret Field data
• Damping mechanisms of internal waves
  Theoretical study
• Summary

4
Seiches in non-rotating lake
ts
h
x
h
u
L
5
Modes
1st mode (r1)
2nd mode (r2)
3rd mode (r3)
Currents
Decomposition into modes
6
Modal analysis
7
Modal analysis
Each mode evolve independently
Spatial
Forcing
Temporal
r 1, 2, 3,
8
Horizontal structure excitation of internal
wave gyres in Lake Biwa

• Understanding relationship between horizontal
  structure of modes their excitation by winds
• Extraction of partitioning of energy input into
  different modes from simulation results

Wind stress
1
1
Temperature
Gyres
Internal waves
density
3
2
BBL
Turbulence
Bottom stress
9
Lake Biwa
Internal waves (BITEX93)
Gyres (Jun 1994)
Counter-clockwise
Clockwise
Velocity 10 15 cm s-1
10
Modes in Lake Biwa
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Excitation - IWs





Wind
-
-
-

NE winds
H1 K dominant
12
Excitation - IWs
-



Wind



-
NE winds
H1 K dominant
NW winds
H1, 2 4 K, H1 P comparable
Winds can preferentially excite certain modes
13
Excitation - Gyres
Wind stress curl
Positive in north
Negative in south
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Numerical Simulation
ADCP
3D hydrodynamic model ELCOM
- Spatially variable wind
- No heat transfer / No inflows outflows
ELCOM
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Energy Partitioning
Strong winds
Excite IWs
but damp quickly
Calm periods
Gyres dominate
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Summary 1

• Winds excite modes that have similar velocity
  pattern in the surface layer
• Internal waves receive large part of energy input
  from winds, and they are damped quickly
• Gyres dominate during the calm periods due to
  slower damping

Published as Shimizu, K., J. Imberger, M.
Kumagai. 2007. Horizontal structure and
excitation of primary motions in a strongly
stratified lake. Limnol. Oceanogr. 52 2641-2655
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Energetics damping of internal waves in Lake
Kinneret

• Estimate damping rates and energetics of internal
  waves from thermistor chain data
• Illustrate effects of internal wave structure on
  near-bottom mass transport

Wind stress
1
1
Temperature
Gyres
Internal waves
density
2
3
BBL
Turbulence
Bottom stress
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Lake Kinneret
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Internal wave modes in Lake Kinneret
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Mode fitting
- Fit five dominant internal waves
Optimize a0(m) Td(m)
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Damping rates energy dissipation
Damping time Td (day)
Period T (hr)
Dissipation rate D (GJ day-1)
Internal wave
V2H3 Kelvin
2.1
1.2
23.3
V1H1 Kelvin
1.7
1.7
22.1
V2H1 Poincare
3.3
0.2
17.3
V1H3 Kelvin
0.83
0.3
10.8
V1H1 Poincare
1.1
0.4
10.0
If internal waves are damped by bottom friction,
Energy dissipation Work done against bottom
friction
Cb 0.0029
D Rate of energy dissipation
Cb Bottom drag coefficient
vb Estimated velocity field
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Near-bottom transport
Current
r
Turbulence
Entrainment of hypolimnetic water
BBL
Resuspension / deposition
Diffusion of soluble materials
Bottom friction
Dissipation rate
Data source Marti Imberger (2006)
Hydrobiologia
Data sourceLemckert et al. (2004) J. Hydraul.
Eng.
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Near-bottom transport
Mass transfer coefficient(BBL-sediment)
Bottom shear stress(resuspension)
Entrainment rate(hypolimnion - BBL)
Lorke et al. (2003) Limnol. Oceanogr.
Gloor et al. (2000) J. Geophys. Res.
Internal wave structure
Large spatial variability
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Summary 2

• Proposed a method that
• extracts energy budgets and damping rates of
  individual internal waves from field data
• estimates spatial variability near-bottom mass
  transport
• 2. Internal waves were damped primarily by bottom
  friction
• 3. Spatial structure of internal waves can cause
  large spatial variability of near-bottom
  transport processes

In press as Shimizu, K. J. Imberger.
Energetics and damping of internal waves in a
strongly stratified lake. Limnol. Oceanogr.
25
Damping mechanisms of internal waves in a
stratified rotating basin

• Understanding how internal waves are damped by
  friction confined in thin bottom boundary layers

Wind stress
1
1
Temperature
Gyres
Internal waves
density
3
2
BBL
Turbulence
Bottom stress
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Simplified problem
Circular basin
Flat bottom, vertical sidewall
Shallow basin (H/Rltlt1)
Linearly stratified
Rotating with frequency 2f
Viscosity is included only in thin boundary
layers
2f
Frequency (compared to f)
Low
High
r
H
Damping mechanism
Spin-down
Wave cancelling
Johns (1968)Dore (1968) Mei Liu (1973)
Barcilon Pedlosky (1967)Greenspan(1968)Walin
et al. (1969)Gill (1982)
2R
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Internal wave cancelling
z
r
Energy flux F pressure p velocity w
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Spin-down
2f
r
H
Ekman layer
Bottom Corner Region
2R
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Bottom boundary layer structure
Ekman layer
t T/4
t T/4
t T/4
t T/4
t 0
t 0
t 0
t 0
w 3f
w 0.1f
w f
w 1.1f
w 0.9f
Direction of boundary layer transport
45o to the left
45o to the right
Parallel
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Damping rates
R1A1 Kelvin
g / 2w
Internal wave cancelling
Spin-down
R1A1 Poincare
g / 2w
Burger number
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Comparison
Circular basin
R 6.4 km
H 30 m
Antenucci Imberger (2001)
2p/f 22.4 hr
S 0.6
n 10-4 m2 s-1
Yeates Imberger (2001)
Period (hr)
Damping time (day)
Field data
Theory
5.8
1.7
22.1
V1H1 Kelvin
2.6
V1H1 Poincare
1.1
10.0
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Summary 3

• Thin boundary layers can cause damping of
  internal waves without diffusion through a
  combination of internal wave cancelling and
  spin-down.
• Predicted and observed damping rates were
  comparable, but theory under-predicted damping
  rates.

Under internal review. To be submitted to J.
Fluid Mech.
33
Conclusions

• Winds excite modes that have similar surface
  velocity structure with the wind stress pattern
• Internal waves receive dominant part of the
  energy input from winds and provide the energy to
  near-bottom mixing
• Gyres are damped slowly and important for
  long-term horizontal transport
• Spatial structure of internal waves can cause
  significant variability of near-bottom mass
  transport processes
• Internal waves are damped quickly by bottom
  friction through internal wave cancelling and
  spin-down

34
Acknowledgements

• Jörg
• Japanese Government Scholarship, Tokyo-Tech
  Long-term study support program CWR ad-hoc
  scholarship
• Lake Biwa Research Institute (M. Kumagai, C.
  Jiao)
• J. Antenucci, C. Dallimore, A. Gomez-Giraldo, T.
  Johnson, P. Okely, T. Shintani
• Officemates P. Yeates, S. Morillo, A. de la
  Funte, C. Boon, P. Okely, P. Huang
• Students staff in CWR SESE
• Family