- 16: Planning, Site , Wed 19-Jul-2000 10:22:56 MDT, CSAT spectral response
Don:
>No strong opinion. If the path averaging limits the response to about 10 Hz,
>there doesn't seem to be much reason to not block average.
The half-power point for w is at k1*d = 4.3 for the CSAT; those for u
and v are around k1*d = 20, although the transfer function for u does
not decrease monotonically with k1 but has a local minimum of 0.65
around k1*d = 3 and a local maximum of 0.9 around k1*d = 9.
The CSAT has a path length of 11.55 cm (the vertical projection is
10 cm), so for the vertical component the half-power point is at a
frequency equal to 5.9*U(m/s).
The wind speed climatology in that area for July 1991 is
Spd.5m (m/s)
1st Qu.:1.472
Median :2.121
Mean :2.292
3rd Qu.:2.912
Max. :7.076
where Qu is Quartile. This puts the w half-power point generally
between 10 and 20 Hz.
Tom
- 12: Planning, Site , Tue 04-Jul-2000 08:17:43 MDT, Revised sonic arrays
I found that the lowest that the sonic could be mounted on the second
section of the PAM mast is 3.25 m. Thus I have revised the arrays as
follows.
Notation:
z = height
1,2 refers to crosswind lines of 9 and 5 sonics, respectively
S = sonic spacing
W = 4*S = width of 5 sonic filter
kc = wave number at filter half-power cutoff
Kc = kc*z/(2*pi)
Array 1: z/w = 0.25, Kc = 0.13
e.g. S1 = 3.25m, z1 = 3.25m; S2 = 6.5m, z2 = 6.5m
|<---------------- 26 m --------------->|
| | | | | | | | |
| | | | | | | | |
x | x | x | x | x -- 6.5m
| | | | | | | | |
x x x x x x x x x -- 3.25m
__/_\__/_\__/_\__/_\__/_\__/_\__/_\__/_\__/_\__
elevation ///////////////////////////////////////////////
xx x xx x xx x xx x xx
|| | || | || | || | ||
|| | || | || | || | ||
|| | || | || | || | ||
/\ /\ /\ /\ /\ /\ /\ /\ /\
plan view /__\ /__\ /__\ /__\ /__\ /__\ /__\ /__\ /__\
* Emulates current LES capabilities.
* Kc near peak of w spectrum only for very strong convection.
* Note that for all arrays, changes in stability change the relation
between Kc and the turbulence spectra.
* Uses nine 10m PAM masts placed foot-to-foot for 3.25m spacing.
Array 2: z/w = 0.5, Kc = 0.27
e.g. S1 = 2.17m, z1 = 4.33m; S2 = 4.33m, z2 = 8.67m
|<----------- 17.3 m ---------->|
x | x | x | x | x -- 8.67m
| | | | | | | | |
| | | | | | | | |
x x x x x x x x x -- 4.33m
| | | | | | | | |
__/_\_/_\_/_\_/_\_/_\_/_\_/_\_/_\_/_\__
elevation ///////////////////////////////////////
xx x xx x xx x xx x xx
|| | || | || | || | ||
|| | || | || | || | ||
|| | || | || | || | ||
/\ __ /\ __ /\ __ /\ __ /\
plan view /__\\ //__\\ //__\\ //__\\ //__\
\/ \/ \/ \/
* Duplication of Tong measurements, but with greater range of stability.
* Kc near peak of w spectrum for convective case.
* Uses nine 10m PAM masts, nested as shown for 2m spacing; 4 sonic
booms will point down when masts are lowered.
Array 3: z/w = 1, Kc = 0.53
e.g. S1 = 2.17m, z1 = 8.67m; S2 = 1.08m, z2 = 4.33m
|<----------- 17.3 m ---------->|
x x x x x x x x x -- 8.67m
| | | | | | | | |
| | | | | | | | |
| | | x x x x x | | | -- 4.33m
| | | | | | | | |
__/_\_/_\_/_\_/_\_/_\_/_\_/_\_/_\_/_\__
elevation ///////////////////////////////////////
x x x x x x x x x x x
| | | | | | | | | | |
| | | | | | | | | | |
| | | | | | | | | | |
/\ __ /\ __ /\ __ /\ __ /\
plan view /__\\ //__\\ //__\\ //__\\ //__\
\/ \/ \/ \/
* Extends range of Kc into Marc's "transition" regime.
* Kc near peak of w spectrum for neutral case.
* Uses nine 10m PAM masts as above, with a triple boom
on the center tower to add two sonics w/S2=1.08m at z2=4.33m.
Array 2.5: z/w = 0.75, Kc = 0.40
e.g. S1 = 2.17m, z1 = 6.5m; S2 = 1.08m, z2 = 3.25m
|<----------- 17.3 m ---------->|
| | | | | | | | |
| | | | | | | | |
x x x x x x x x x -- 6.5m
| | | | | | | | |
| | | x x x x x | | | -- 3.25m
__/_\_/_\_/_\_/_\_/_\_/_\_/_\_/_\_/_\__
elevation ///////////////////////////////////////
x x x x x x x x x x x
| | | | | | | | | | |
| | | | | | | | | | |
| | | | | | | | | | |
/\ __ /\ __ /\ __ /\ __ /\
plan view /__\\ //__\\ //__\\ //__\\ //__\
\/ \/ \/ \/
* Alternative to arrays 2 and 3 if time is short
* Uses nine 10m PAM masts as above, with a triple boom
on the center tower to add two sonics w/S2=1.08m at z2=3.25m.
Array 4: z/w = 2, Kc = 1.1
e.g. S1 = 0.5m, z1 = 4m; S2 = 0.63m, z2 = 5m
|<----- 4m ---->|
||======x=x=x=x=x======|| -- 5m
||==x=x=x=x=x=x=x=x=x==|| -- 4m
|| ||
__||_____________________||__
elevation /////////////////////////////
x x x x x x x x x
| | | | | | | | |
| | | | | | | | |
_____|_|_|_|_|_|_|_|_|_____
/_\_/_\_/_\_/_\_/_\_/_\_/_\
plan view /\ /\
/__\ /__\
* Kc in inertial range for convective case, near peak of w spectrum
for stable case.
* Requires very different support structure than arrays 1-3.
* Increased possibilty of flow distortion due to small spacing;
we can check for flow distortion in post-analysis by comparing
time-averaged turbulence statistics for all sonics.
This intercomparison can be done as a function of S.
* Setting up this array may require about 2 days of effort,
but the support structure can perhaps be set up while
arrays 2&3 are collecting data.
* It's possible that we might run out to time before using this array,
depending on how quickly we complete data collection with previous
arrays. Note also that the wind climatology may not be as favorable
toward the end of September.
* Uses four 5.5m ASTER steel tower sections.
- 8: Planning, Site , Mon 31-Jan-2000 09:38:01 MST, Proposed sonic arrays
In order to attempt to bring closure to the experimental design, I
propose the following possible set of sonic arrays. They are listed in
order of priority, so that we will collect data with each array until
we have a data set satisfying some (yet to be specified) criteria.
Then we will proceed to the next array. I have changed the order of
the arrays. This order is perhaps slightly more logical from a
logistical standpoint and proceeds systematically from large spacings
and small Kc to small spacings and large Kc.
Notation:
z = height
1,2 refers to crosswind lines of 9 and 5 sonics, respectively
S = sonic spacing
W = 4*S = width of 5 sonic filter
kc = wave number at filter half-power cutoff
Kc = kc*z/(2*pi)
Array 1: z/w = 0.25, Kc = 0.13
e.g. S1 = 3m, z1 = 3m; S2 = 6m, z2 = 6m
|<---------------- 24 m --------------->|
| | | | | | | | |
| | | | | | | | |
x | x | x | x | x -- 6m
| | | | | | | | |
x x x x x x x x x -- 3m
__/_\__/_\__/_\__/_\__/_\__/_\__/_\__/_\__/_\__
elevation ///////////////////////////////////////////////
xx x xx x xx x xx x xx
|| | || | || | || | ||
|| | || | || | || | ||
|| | || | || | || | ||
/\ /\ /\ /\ /\ /\ /\ /\ /\
plan view /__\ /__\ /__\ /__\ /__\ /__\ /__\ /__\ /__\
* Emulates current LES capabilities.
* Kc near peak of w spectrum only for very strong convection.
* Note that for all arrays, changes in stability change the relation
between Kc and the turbulence spectra.
# Uses nine 10m PAM masts placed foot-to-foot for 3m spacing.
Array 2: z/w = 0.5, Kc = 0.27
e.g. S1 = 2m, z1 = 4m; S2 = 4m, z2 = 8m
|<------------ 16 m ----------->|
x | x | x | x | x -- 8m
| | | | | | | | |
| | | | | | | | |
x x x x x x x x x -- 4m
| | | | | | | | |
__/_\_/_\_/_\_/_\_/_\_/_\_/_\_/_\_/_\__
elevation ///////////////////////////////////////
xx x xx x xx x xx x xx
|| | || | || | || | ||
|| | || | || | || | ||
|| | || | || | || | ||
/\ __ /\ __ /\ __ /\ __ /\
plan view /__\\ //__\\ //__\\ //__\\ //__\
\/ \/ \/ \/
* Duplication of Tong measurements, but with greater range of stability.
* Kc near peak of w spectrum for convective case.
# Uses nine 10m PAM masts, nested as shown for 2m spacing; 4 sonic
booms will point down when masts are lowered.
Array 3: z/w = 1, Kc = 0.53
e.g. S1 = 2m, z1 = 8m; S2 = 1m, z2 = 4m
|<------------ 16 m ----------->|
x x x x x x x x x -- 8m
| | | | | | | | |
| | | | | | | | |
| | | x x x x x | | | -- 4m
| | | | | | | | |
__/_\_/_\_/_\_/_\_/_\_/_\_/_\_/_\_/_\__
elevation ///////////////////////////////////////
x x x x x x x x x x x
| | | | | | | | | | |
| | | | | | | | | | |
| | | | | | | | | | |
/\ __ /\ __ /\ __ /\ __ /\
plan view /__\\ //__\\ //__\\ //__\\ //__\
\/ \/ \/ \/
* Extends range of Kc into Marc's "transition" regime.
* Kc near peak of w spectrum for neutral case.
# Uses nine 10m PAM masts as above, plus either two shorter 4-5m masts
or perhaps some double booms to add two sonics w/S2=1m at z2=4m.
Array 4: z/w = 2, Kc = 1.1
e.g. S1 = 0.5m, z1 = 4m; S2 = 0.63m, z2 = 5m
|<----- 4m ---->|
||======x=x=x=x=x======|| -- 5m
||==x=x=x=x=x=x=x=x=x==|| -- 4m
|| ||
__||_____________________||__
elevation /////////////////////////////
x x x x x x x x x
| | | | | | | | |
| | | | | | | | |
_____|_|_|_|_|_|_|_|_|_____
/_\_/_\_/_\_/_\_/_\_/_\_/_\
plan view /\ /\
/__\ /__\
* Kc in inertial range for convective case, near peak of w spectrum
for stable case.
* Requires very different support structure than arrays 1-3.
* Increased possibilty of flow distortion due to small spacing;
we can check for flow distortion in post-analysis by comparing
time-averaged turbulence statistics for all sonics.
This intercomparison can be done as a function of S.
* Setting up this array may require about 2 days of effort,
but the support structure can perhaps be set up while
arrays 2&3 are collecting data.
* It's possible that we might run out to time before using this array,
depending on how quickly we complete data collection with previous
arrays. Note also that the wind climatology may not be as favorable
toward the end of September.
# Uses four (two steel, two Al) 5.5m ASTER tower sections.
- 7: Planning, Site , Wed 19-Jan-2000 13:30:16 MST, eddy decay time scale
Tom,
Sorry I didn't get over yesterday. Below are my thoughts on the time scale.
I propose to use what we had in the footprint paper (1992):
\tau = (2/C_0) \sigma_w^2/\epsilon
although there are some uncertainties in C_0 (3 to 5) and whether or not
one uses \sigma_w for the u and v velocity components. Use of \sigma_w
would result in the smallest time scale and perhaps that is what should
be used for estimation purposes, i.e., to see if Taylor's hypothesis holds.
If you use \sigma_w = 1.3 u_\ast and \epsilon = u_\ast^3/(k z), then the
\tau = (1.35/C_0) z/u_\ast.
For C_0 = 3, 4, and 5, the coeficient c_1 = 1.35/C_0 = 0.45, 0.34, and 0.27,
respectively. I would probably just choose 0.34 for now and see what you
get. We can talk about this when you wish. But I guess, you probably need
to make some estimates soon about the validity of Taylor's hypothesis.
(Sorry I didn't get this to you much earlier. I have been tied up in getting
a talk ready for the AMS Air Pollution Conference in Long Beach, CA last week,
going to and chairing the Conference. Everything went well.)
Jeff
- 6: Planning, Site , Wed 19-Jan-2000 13:29:14 MST, Non-normal wind direction
If the wind deviates from normal to the array by D degrees, then
we will need to lag the data by a time up to 4S sin(D)/U. Here S
is sonic spacing and I assume a 5 sonic resolved-scale filter.
Jeff estimates the time scale for decay of an eddy to be 0.34 z/u*.
Thus we want
4S sin(D)/U << 0.34 z/u* or
sin(D) << 0.34 z/W U/u*
where W=4S is the width of the spatial filter.
We have been considering 0.13 < z/W < 0.6. In the worst case,
sind(D) << 0.34*0.13/0.1 = 0.44
or sin(D) ~ 0.04 => D ~ 2.5 degrees!
In the best case,
sind(D) << 0.34*0.6/0.1 = 2
or sin(D) ~ 0.2 => D ~ 12 degrees
This is not too encouraging.
- 5: Planning, Site none, Fri 17-Dec-1999 12:19:45 MST, PAM mast packing
How closely can we "pack" PAM tripods for SGS?
The PAM tripods have an equilateral triangle footprint, L=115" on each
side. Thus if we place them side-by-side in the same orientation, the
spacing of the masts would be L = 2.92m.
/\ /\ /\ /\ /\
/__\/__\/__\/__\/__\
However the mast itself is only a distance of L/3 from one side of the
tripod footprint, so alternating the orientations of the tripods by
180 degrees allows a spacing of the masts equal to 2/3 L = 1.95m.
/\ __ /\ __ /\
/__\\ //__\\ //__\
\/ \/
Note that the tripods are also staggered in order to line up the masts
in a straight line. The tripods could perhaps be nested even a little
closer, since the only structure on the outer edges of the footprint is
the three legs themselves.
The PAM tripod structure without the electronics box (recall that we
will be using ADAMs to ingest the sonic data) are actually very open,
so I don't expect the flow distortion with PAM masts to be any worse
than with some sort of horizontal beam to support the sonics.
- 4: Planning, Site none, Fri 17-Dec-1999 09:00:13 MST, calculating spatial averages
How to calculate 2D spatial averages ala Tong et al?
The devil is in the details!
Let
w = arbitrary flow variable, e.g u,v,w, etc,
number of sensors in single crosswind filter = 2n+1,
sensor spacing = S,
mean wind speed = U,
mean wind direction (relative to array normal) = D,
with D defined positive for wind blowing from x_{-n} to x_n
Then effective sensor spacing = S' = S cos(D)
Crosswind average =
= Sum_{j=-n}^{j=n} C_j w(x_j,t+jdt)
where dt = S sin(D)/u with u the instantaneous wind speed.
Does Taylor's hypothesis need to hold over the time +/-n dt required
for a single filter operation (or is it 2n dt?) or the time +/-2n dt
required for a second order filter (or is it 4n dt?)? I have my own
opinion, but would like some independent assessments.
Streamwise average =
w~(t) = 1/(2T) Int_{-T}^T C(t') w(t+t') dt'
where T = nS cos(D)/U gives a streamwise filter with the same
width, 2UT = 2nS', as the crosswind filter.
C_j and C(t') are weighting functions used to define filter shape:
Sum_{j=-n}^{j=n} C_j = 1
1/(2T) Int_{-T}^T C(t') dt' = 1
Then the resolved and sgs flow variables are
w^r = ~
w' = w - w^r
Note that both w^r and w' remain time-dependent, stochastic variables.
It would appear to be straightforward to apply this process again to
obtain second moments such as
~ = < ~ (w - ~) >~
This is all fine for a stationary flow field. But how do we define U
and D for a non-stationary flow as will be the case for much of the
data? We need to define or simply choose a procedure for our real-time
data analysis in the field. Do we just pick an arbitrary averaging
time, such as 5, 15, 30, or 60 minutes? There is perhaps some logic to
make this averaging time on the order of T = nS'/U. But note the
difficulty of calculating the latter if U=f(T).
- 3: Planning, Site none, Fri 03-Dec-1999 14:35:32 MST, sonic spacing
12/3/99
I have looked briefly at the issue of sonic spacing:
For a height of z=6m and a filter width of W=12m=(4*3m), Tong et al
calculate a filter cut-off at a wavelength lc = 2*pi*z/1.67.
Thus lc = 2*pi*(6m)/1.67 = 22.6m = 1.88*W,
where W = width of spatial filter (of Tong design)
We want lc to match lw, the wavelength at the peak in the vertical
velocity power spectrum.
From Kaimal and Finnigan, Fig. 2.9, this peak occurs
at:
z/L z/lw W(z=6m)
--- ---- -------
-1 0.2 16m
-0.5 0.3 11m
0 0.6 5.3m
0.5 1 3.2m
1 1.5 2.1m
Note that the peak wavelength of the u spectrum is 5-10 times larger
than that for the w spectrum.
Combining these two, W = z/(1.88*z/lw), which is in the 3rd column
above.
Thus it would appear that we might choose perhaps 12m (Tong's spacing),
6m and 3m for the filter width (sonic spacing of 3m, 1.5m and 0.75m!)
in three different arrays that cover the range of stability. Note
that the closer spacings would require wind directions very close to
normal to the array in order to minimize flow distortion of one sonic
by another. The spacing is proportional to z, so the higher we can
place the array, the better.
Comments? I infer that practical LES simulations do not use filter
widths as small as 3m, so perhaps we do not want to use lw as the
criterion for stable conditions? I recall that Chin-Hoh and Peter
wanted to use perhaps 12m and 24m as two filter widths. Obviously this
is not compatible with the lw criterion, which was not apparent to me
at our meeting.
12/17/99
Peter suggests that we investigate two cases,
Kc = z/lc = 0.067 and 0.333.
Kc = z/(1.88*W) = z/(7.5*S) , where S = sonic spacing
Thus for Kc = 0.067: S/z = 2 and a height of 5m implies S = 10m
Kc = 0.333: S/z = 0.4 and a height of 5m implies S = 2m
Could we keep the spacing the same and just change the height?
Let S=4m, then for Kc = 0.067: z = S/2 = 2m
for Kc = 0.333: z = 2.5S = 10m
It's almost doable!
We could get Kc as high as z/(7.5*S) = 9m/(7.5*2m) = 0.6
- 2: Planning, Site none, Fri 03-Dec-1999 14:32:20 MST, Taylor's hypothesis
As Don noted during our meeting, the validity of Taylor's hypothesis
depends on sigw/U << 1 and sigu/U << 1.
sigw/U = sigw/u* x u*/U = sigw/u* x k/[ln(z/zo - psi(z/L)]
sigu/U = sigu/u* x u*/U = sigu/u* x k/[ln(z/zo - psi(z/L)]
sigw/u* ~ 1.3, sigu/u* ~2, so the validity of Taylor's hypothesis
depends more critically on sigu/u*. Note that there is no explicit
dependence on wind speed and only a weak dependence on stability.
For neutral stability, z=6m and zo=3cm,
sigu/U = 2*0.4/ln(6/0.03) = 0.15
I have plotted sigma_u/U versus U for the SJVAQS field program, held in
the San Joaquin valley of CA in 1991. For 5m wind speeds above 3 m/s,
sigma_u/U ~ 0.2. At lower wind speeds, the range of sigma_u/U
increases with decreasing wind speed, from values as small as 0.05
around 2 m/s up to values approaching 1 below 1 m/s.
Although the possible values for sigma_u/U increase at low wind speeds,
there is still plenty of data at low wind speeds with values equal to
or below 0.2. The values below 0.2 are associated with low values of
u*/U. These occur during the stably-stratified conditions (at night)
when the drag coefficient is low. Thus I anticipate no problem with
satisfying Taylor's hypothesis if sigma_u/U < 0.2 is satisfactory.