Ninni Liukko&Timo Huttula
20.1.2008
SCALES OF TURBULENCE IN TWO DIFFERENT CASES
CASE DESCRIPTION
1. Poronselkä, Lake Päijänne 17.8.2007
- ADCP current data (three observations from each depth cell in about 30 s time, 17
(14) 2m thick depth cells)
- CTD data after couple of hours
- Thermocline about in depth of 11-21m
Poronselkä 17.8.2007
Density kg/dm ^3
0.9975
0.998
0.9985
0.999
0.9995
1
0
5
Depth (m)
10
15
20
25
30
35
40
Poronselkä 17.8.2007
Velocity (m m /s)
0
0
5
Depth (m)
10
15
20
25
30
35
40
50
100
150
200
250
2. Lake Vesijärvi 22.8.2006
- ADCP current data (12 observations from each depth cell in about 3 min time, 17
1m thick depth cells, average error velocity below 1,5 cm/s in all cells except for
depths 11m and 17m)
- CTD data from a site nearby (about 200m away, 30 min before ADCP
measurement)
- Thermocline about in depth of 8-12m
Vesijärvi C10 22.8.2006
Density (kg/m 3)
998
998.2
998.4
998.6
998.8
999
999.2
999.4
999.6
0
2
4
Depth (m)
6
8
10
12
14
16
18
20
Vesijärvi 22.8.2006
Velocity (m m /s)
0
0
2
Depth (m)
4
6
8
10
12
14
16
18
20
20
40
60
80
100
120
140
RESULTS
Brunt-Vaisala frequency, Period length
Brunt-Vaisala frequency or buoyancy frequency (N) was calculated over the thermocline
in both cases. In Poronselkä, 2007, N was 0.041 rad s-1, which means a period length (Pw)
of 154 s = 2 min 34 s. In Lake Vesijärvi, 2006, N was 0.047 rad s-1 and the period length
was 134 s = 2 min 14 s.
The buoyancy frequency is the frequency of the oscillation that results when the density
interface is displaced and then left to return to its rest position. The oscillation spreads out
as a moving internal wave. The periof of about two minutes means an atypically high
gradient and fast wave when compared to the ocean, where the period of waves may be
from 10 minutes to several hours (Mann & Lazier 1991).
Richardson number
The Richardson number estimates the likelihood that internal waves in a density interface
will become unstable and break up into turbulence. Ri is a ratio between the buoyancy
forces (which tends to limit the amplitude of the waves) and the shear force (which tend
to increase the size of the waves). If this ratio is greater than 0.25 waves of all
wavelengths are stable (Turner, 1973).
The Richardson number calculated over thermocline was 388 in Poronselkä (2007) and
172 in Vesijärvi (2006). This suggests that the stratification in both cases was strong and
that internal waves were not breaking in the thermocline.
Approximation of friction velocity
Friction velocity (u*) for the thermocline were approximated so that u*≈le(du/dz), where
le is the length scale and du/dz is the velocity gradient. Calculation suggest that friction
velocity in thermocline was 0.02 m/s in Poronselkä and 0.01 m/s in Lake Vesijärvi. This
result is calculated to cover the whole thermocline depth (10 m in Poronselkä and 4m in
Lake Vesijärvi) and is therefore only an indicative value of the real friction velocity,
which may also change a lot in the thermocline depth.
To approximate more accurately the friction velocity more velocity and measurements
should be done with small depth intervals.
Dissipation of kinetic energy
Dissipation of kinetic energy (E) through thermocline were calculated with the
approximated value of friction velocity: E= (u*)² (du/dz). E was found to be 8.8*10-7 m2s-
3
for the Poronselkä case and 7.2*10-7 m2s-3 for the Lake Vesijärvi case. These values are
in the range of suggested values of E in the ocean (10-6 – 10-9 Mann & Lazier 1991).
Buoyancy length scale
The buoyancy length scale (Lb) is the size of the largest turbulent eddies. The largest
eddies occur when the inertial forces associated with the turbulence are about equal to the
buoyancy forces. This size is estimated from the turbulent energy dissipation rate E and
the buoyancy frequency N by Gargett et al. (1984): Lb=(E/N³)1/2.
The buoyancy length scale was calculated to be 0.11m in the Poronselkä case and 0.08m
in the Lake Vesijärvi case. These values are quite small when compared to the ocean,
where the buoyancy length scale may be about 10 m in the mixed layer and 1 m in the
deep ocean or in stratified regions (Mann &Lazier p. 20).
Batchelor scale for heat
Batchelor scale is the length scale of the smallest fluctuation of any property of diffusion
constant D and it is given by Ld = 2П(vD2/E)1/4.
In Poronselkä case, Ld was found to be about 2.5 mm and in Lake Vesijärvi case 2.6 mm.
These values were calculated with the approximated values of E and they are in the range
of values for smallest temperature fluctuations (2-13mm) suggested by Mann & Lazier
(1991).
Kolmogorov scales
The Kolmogorov length scale Lm was approximated to be 1 mm in both cases. The
Kolmogorov time scale was about 1 s in both cases. The Kolmogorov velocity scale was
about 1 mm/s in both cases. The approximated kinetic energy dissipation value (E) was
used in these approximations. Because E was not estimated from the finest scales of the
velocity gradient but from velocity averages from 1-2 meters depth intervals, the
approximations of the Kolmogorov scales may be unrealistic.
Reynolds number
Reynolds number (Re) was calculated for thermocline in both cases by Re=ul/v, where u
is the average velocity for thermocline, l is the thermocline depth and v is the kinematic
viscosity. In Poronselkä case Re was found to be 817333 and in Lake Vesijärvi case
444200. These values suggest that the flow was turbulent in the thermocline in both cases,
because Re>2000.
Wedderburn number
The intensity of stratification, mixing, and the influence of wind stress can be examined
by the calculation of the dimensionless Wedderburn number (Monismith, 1986):
W = g'h2/(u*2Lm)
The formula combines mixing depth h (in m), the reduced acceleration of gravity g' (m s2
), proportional to the density jump across the thermocline, shear velocity u* (m.s-1), and
lake length Lm (m), with
g' = 2g(r2-r1)/(r2+r1)
r1, r2 = densities of epilimnion and hypolimnion respectively
g = acceleration of gravity, 9.8 m.s-2
u*2 = (rair/rwater)CV2 = 0.0011 * 0.0014 * V2
C = coefficient of drag
V = wind speed
The Wedderburn number was calculated for Poronselkä and Lake Vesijärvi cases with
lake length of 5000m. Both open lake areas are about 5 km in longitudinal direction.
Wind speed was expected to be 5 m/s in both cases. W was found to be about 12 in
Poronselkä case and 3 in Lake Vesijärvi case. These high values of W indicate stability
and a well developed stratification, when low numbers of W (particularly for W<0.5)
indicate a strong shear stress at the thermocline, tilting of the thermocline with related
mixing and possible upwelling at the windward end of the lake.
Froude number
Froude number (Fr) were calculated for the epilimnion and found to be about 0.01 in both
cases. This means that the flow in the epilimnion were subcritical (true, when Fr<1) in
both cases.
Scale for vertical convection
Scale for vertical convection (Uf) can be calculated by
Uf=((αghH)/(Cpp0))1/3,
where α is coefficient of heat expansion, g is acceleration due to gravity, h is epilimnion
depth, H is heat flux from water, Cp is capasity of specific heat for water and p0 is
epilimnion density (Fischer et al. 1979).
When H is 300 W m-2, Uf was found to be about 1 cm s-1 in both cases. With H=30 Uf
was 0.5 cm s-1.
Gargett A.E., Osborn T.R. & Nasmyth P.W. 1984. Local isotropy and the decay of
turbulence in a stratified fluid. J. Fluid. Mech. 144:231-280 (ei saatavissa??)
Monismith, S., 1986. An experimental study of the upwelling
response of stratified reservoirs to surface shear stress.
J. Fluid Mech., 171: 407-439. (ei saatavissa??)
Fischer et al. 1979. Mixing in inland and coastal waters. Kirja.