Mixed Layer Growth

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Weather/Wind Forecasting and
Impact of Surface Conditions on
Vertical Profile of Horizontal
Winds
William A. Gallus, Jr.
Dept. of Geological and Atmospheric
Sciences
May 31, 2013
How might you forecast the
weather (or wind in particular)?
Several Methods can be used:
• Climatology – just assume winds will
behave like they normally do at this
location, time of year, and time of day
• Statistical – fit observed data using
multiple regression, automated neural
networks, etc.
• Numerical models (Numerical Weather
Prediction)
Climatology
• Ramp events, very difficult to forecast, can
sometimes be best predicted using
climatology.
• Winds tend to follow seasonal and diurnal
trends (reasons to be explained later) –>
near the ground they are strongest during
the day and weakest at night. Several
hundred meters up, behavior can be
opposite. Not fully understood which way
they will behave around turbine height of
80 m
Both types of
ramps show two
peaks during the
year. One in
May/June
corresponds to
active period for
Low-Level jets and
also thunderstorms.
Other is in Dec/Jan.
Ramp ups increase
sharply to peak in
evening (6-9 pm)
with minimum in
early morning (6-9
am).
Ramp downs have
peak in early
morning (6-9 am)
with very pronounced
minimum in mid-day
hours (9am-3pm).
Statistical Approaches
• Can be applied to both observations and
model output to look for
relationships/correlations. Doesn’t have to
make obvious sense. You might find that
a prediction of wind 2 hrs from now is most
accurate if an equation is used based on
phase of the moon, score of recent Cubs
game, etc!
Statistical Approaches
• Traditionally, in operational forecasting,
MOS (Model Output Statistics) is used to
refine NWP model output to improve
predictions of surface temperature, dew
point, and winds. Otherwise, statistical
approaches are often avoided by scientists
as being “black box”, but they can yield the
very best forecasts. These approaches,
e.g. MOS, can make use of the other 2:
raw NWP output, climatological trends
Numerical Weather Prediction
Models
• How can you use equations to predict the
weather?
Numerical Weather Prediction
Models
• How can you use equations to predict the
weather?
• If you have an equation that involves a
d/dt, like du/dt = A + B + C, then you can
predict the future (you know how the
variable, like wind, changes with respect to
time)
Can you think of an equation
that would allow us to predict
future wind?
Can you think of an equation
that would allow us to predict
future wind?
F = ma
a = dV/dt
So dV/dt = F/m
How do you think we can solve such an equation?
Numerical Weather Prediction
Models
• Weather forecasting models are generally
built on the Navier-Stokes equations,
which come from the basic laws of
physics.
• These equations are usually discretized,
typically via finite differencing, and
because they are prognostic equations,
they can then predict the future (one time
step at a time)
• For wind speed prediction, the forecasting
equations of interest derive from F=ma.
We can re-arrange to write a=F/m, which can yield
component equations:
Du/Dt = F/m
Dv/Dt = F/m
Dw/Dt = F/m
But, we usually like to forecast at a fixed point
(Eulerian framework) instead of in a Lagrangian
sense, so we expand the total derivative
• We thus have equations that look like:
∂u/∂t = -v∇u - w∂u/∂z = F/m
• For horizontal winds (u and v), the forces
of importance on the right-hand side
include pressure-gradient force, Coriolis
force, and Friction. The friction
contribution is mostly from the ground.
• Pressure gradients are the one force that
really “creates” wind. Temperature
changes can lead to pressure changes,
and thus affect the winds
• Coriolis Force (f k x V) just changes the
direction of wind (directed to the right in
Northern Hemisphere)
• Friction ends up being a challenge to treat
in the models
Other considerations for NWP
• Where do you make your predictions
(which kind of horizontal and vertical grid
do you use)?
w
T,u,v
T,u,v
T
v
T
T,u,v
u
T
u
v
w
T
• How exactly do you “discretize” the
equations?
ut+1 = ut + ▲t – ut (ut, i+1- ut, i-1) /▲x - …
ut+1 = ut-1 + 2▲t – (ut, i+1- ut, i) /▲x …
or
• How do you treat atmospheric processes
that are too small to be depicted on your
grid?
Parameterizations are used – and these can
vary greatly. Often used for convection,
microphysics (cloud physics), turbulence,
radiation, land-surface interaction, soil
processes….
• When we solve equations like these in a
numerical forecast model, however, we
are really solving for averages (both
spatial and temporal)
Reynolds Averaging
• Any variable b at any point in space-time
can be expressed as a mean and a
perturbation: b= b + b’
• The average of perturbations is 0 = b’
• Statistics are stationary over averaging
interval
So, if we apply these rules, our equation for
u-momentum looks like:
∂u/∂t + ∂u’/∂t = -(u∂u/∂x + u’∂u/∂x +u∂u’/∂x
+u’∂u’/∂x) – v…..
But averages of single primed terms are 0,
so this becomes…
∂u/∂t = -(u∂u/∂x + u’∂u’/∂x) – (v….
Here, the first term is advection of mean
wind by mean wind.
• Second term is net (mean) contribution
from advection of wind fluctuations by the
fluctuating part of the wind. These
fluctuations are often referred to as
“turbulence” but this is not strictly correct –
they include ALL fluctuations on scales
smaller than the grid volume.
• If we look at the sub-grid scale transport
terms in more detail, we can use the
product rule of differentiation and
rearrange to show that
• ∂u/∂t = -u∂u/∂x -∂/∂x(u’u’) - u’∂u’/∂x
Where the second term is recognizable as
“turbulence flux divergence” (or sub-grid)
Last term can be re-written and shown to
equal zero assuming incompressible
continuity
Closure Issue
• Our entire set of governing equations then
ends up consisting of 7 equations, with 7
first-moment variables (u,v,w,Θ,ρ,p,T – all
of these having overbars on them), but we
have ended up with 8 second-moment
unknowns from the averaging procedure.
• If we make more predictive equations for
the second moments, we don’t fix the
problem, because we then end up with
additional terms involving the third
moment [like ∂/∂x(u’u’u’) ].
• We cannot close the system by adding
equations for the higher-order terms. To
obtain a closed system, we have to do
something about the higher-order terms.
Obtaining a closed system
• We can either ignore higher-order terms
• Or…represent them as a function of lowerorder variables
First-order closure
• For first-order closure, used in almost all
forecasting models, we have equations to
predict the first-order moments, and we
represent higher-order moments as a
function of the first-order ones.
Gradient-transfer theory
• Most common approach to first-order
closure is called gradient-transfer theory.
We represent fluxes as proportional to
gradients of mean variables:
u’b’ = -K ∂b/∂x
So, subgrid flux in a given direction is
proportional to gradient in that direction,
and transport is down-gradient (from high
values of b toward low ones)
• Also, transport depends on the LOCAL
GRADIENT (∂b/∂x)
• The K terms can be anisotropic, meaning
we could have different values for
horizontal versus vertical transport
Specification of Ks
• We can use profile methods – specify
some function giving realistic distribution
of K with height, so that it is small near
ground, large in lower half of Planetary
Boundary Layer (PBL), and smaller again
near top of PBL
• Local stability/mixing length – K’s are
specified by some functional form so they
are small where atmosphere is stable and
increase as atmosphere becomes less
stable
Mixing Length (cont)
• These schemes specify a “mixing length”.
A “law of the wall” type approach is used
so that l -> 0 at the ground (l=kz) where
k=von Karman’s constant (around 0.4).
An upper bound is placed on k so that l< l0
of roughly 50-100 m
• A widely-recognized deficiency in gradienttransfer theory is that sometimes fluxes do
not depend on local gradients. Convective
eddies in daytime PBL transport heat
across the entire PBL depth (from ground
to top), and this transport is related more
to bulk PBL properties than local
gradients. It is possible to try to include
some of these non-local effects.
PBL (Mixed Layer) Growth
• Definition – layer of the atmosphere having its
behavior directly affected by the surface
• The PBL typically grows deeper during the day as
the sun warms the ground, and rapidly collapses at
night
Potential temperature is a concept often used
to depict the PBL –> θ = T (1000/p)R/cp
β=∂θ/∂z
zi
Z
PBL
heat
The layer with uniform potential temperature is referred to
as the mixed layer (or PBL), and its top is usually denoted
zi (where “i” stands for “inversion”)
• The redistribution of air parcels corresponds
to a heat flux. We represent the kinematic
heat flux as w’’θ’’, i.e., correlations of
turbulent vertical motions with turbulent
fluctuations of temperature.
• The heat flux warms the temperature and
deepens the PBL
• A deeper PBL generally means that higherspeed winds from higher levels in the
atmosphere are “mixed” down to the region
near the surface (and vice-versa… frictionslowed air is mixed upward)
Land use/characteristics impact
the PBL
• How much friction is a function of the
characteristics of the earth’s surface
• Lots of trees/hills/tall buildings lead to lots
of friction and slowing of flow
• Cold water or snow prevents warming and
thermals, confining friction to layer right
near ground, without much PBL
development
• Things get really interesting as the sun is
getting ready to set
• Cooling first happens near the ground, so
a stable layer forms there, and this
“decouples” the atmosphere from the
surface. Friction now has very little impact
on the air in the lower atmosphere.
Remnant PBL
Stable layer
• Removal of friction allows the air a short
distance above the ground (50-1000 m) to
begin accelerating, with speeds increasing
and directions veering (clockwise)
• Coriolis force leads to an “inertial
oscillation” so that winds peak in speed
around 1 or 2 am => Low Level Jet
PBL parameterizations in
models – example from WRF
• Non-local:
1) ACM2 (or Pleim)
2) YSU
• Local:
1) MYJ
2) QNSE
3) MYNN2.5
4) MYNN3.0
Biases in 80m wind forecasts using different PBL schemes in WRF model
Note: YSU is most different
Composites of PBL biases by hour. Each line represents a different PBL scheme;
MYJ (Black), MYNN 2.5 (Red), MYNN 3.0 (Blue), Pleim or ACM2 (Green), QNSE
(Cyan), and YSU (Magenta).
Differences in predictions of Low-Level Jets from one case using WRF model
Observed peak
Wind speed as a function of height profile during the LLJ peak on March 24, 2009
at 10pm LST. Black dots represent observations from the 915-mHz wind profiler
while each line represents a different PBL scheme; MYJ (Black), MYNN 2.5 (Cyan),
MYNN 3.0 (Magenta), Pleim or ACM2 (Red), QNSE (Blue), and YSU (Green).
Three hour averaged diurnal cycle of ramp-up events using the
midpoint of the ramp event. Black line is observed ramp-up events.
Note the very different behavior of the YSU scheme, and to a lesser
extent the Pleim
Example of challenges of
creating a good NWP forecast
system for Wind
Based on M.S. work of Adam
Deppe (Deppe et al. 2013 Wea.
Forecasting paper)
NWP Model (WRF) was run over these 2 domains/grid spacings
Which would you think should give the best forecast and why???
Model domain and location of verification data
Figure 2. (Left) The 10 km domain with outline of Pomeroy wind farm (right)
where the individual wind turbines are the black dots and the 80 m
meteorological tower (observed data location) is the X.
Table 2. MAE associated with six PBL schemes using the 00 UTC time initialization and the
GFS ILBC from 10 cases in January 2010. The six member ensemble average and the standard
deviation (measure of model spread) are also listed.
PBL Scheme
MAE ( m s-1)
MYJ
1.38
-1
(ms )
MYNN 2.5
1.43
-1
(ms )
MYNN 3.0
1.38
-1
(ms )
Pleim
1.29
-1
(ms )
QNSE
1.39
-1
(ms )
YSU
1.31
-1
(ms )
Ensemble
1.26
-1
(ms )
Note the ensemble (average) does the best
Standard
Deviation
------------0.66
Table 3. MAE associated with three different GFS perturbations using the YSU and MYNN3.0
PBL schemes from 10 cases in January 2010. The two member ensemble average and the
standard deviation (measure of model spread) are also listed.
Perturbation
Number
MYNN 3.0 MAE
( m s-1)
YSU MAE
( m s-1)
Ensemble MAE
( m s-1)
2
4
15
Standard
Deviation
1.88
1.73
1.80
---
1.60
1.59
1.72
---
1.58
1.53
1.62
0.98
The spread (variety) has increased but not the skill (MAE not
reduced) so this is not a good solution
Table 4. MAE associated with the wind speed at 80 m from two different grid spacings (4 km
and 10 km) from 10 cases in January 2010. The two member ensemble average is also listed.
Grid Spacing
MYNN 3.0 MAE
( m s-1)
YSU MAE
( m s-1)
Ensemble MAE
( m s-1)
10 km
1.37
1.29
1.18
4 km
1.70
1.33
1.27
For this domain and cases, the 4 km model
domain did not work as well. Ideas why?
Table 5. MAE associated with the wind speed at 80 m from three different initialization times
from 10 cases in January 2010. The two member ensemble average and the standard deviation
(measure of model spread) are also listed.
Time Initialization
MYNN 3.0 MAE
( m s-1)
YSU MAE
( m s-1)
Ensemble MAE
( m s-1)
18 UTC
00 UTC
06 UTC
Standard
Deviation
1.42
1.37
1.38
---
1.32
1.29
1.61
---
1.23
1.18
1.28
1.09
Different time initializations lead to lower skill AND higher spread
1
2
3
4
5
6
7
Table 6. MAE calculated for the first 24 hour period. The three PBL schemes with the lowest
MAE were chosen, making up the Day 2
selected ensemble. Times selected indicate s the
number of times a model was chosen as a member of the Day 2
selected ensemble. The nonselected ensemble incorporated the least accurate models for the first 24 hour period. Day 2 All
Member Ensemble incorporated all six model members.
Model Number
00 UTC MYJ GFS with a 10 km
grid spacing
00 UTC MYJ NAM with a 10 km
grid spacing
00 UTC Pleim NAM with a 10 km
grid spacing
00 UTC Pleim GFS with a 10 km
grid spacing
00 UTC YSU NAM with a 10 km
grid spacing
00 UTC YSU GFS with a 10 km
grid spacing
Day 1 MAE ( m s-1)
Times selected
2.51
5
2.61
6
2.58
4
2.36
9
2.32
11
2.37
10
Ensemble Mean
1.97
Day 2 selected ensemble best
MAE
5/15
Day 2 non-selected
ensemble best MAE
4/15
Day 2 All Member
Ensemble best MAE
6/15
1
Table 7. MAE for wind speed at 80 m associated with the neighborhood approach.
Grid
Averaging
MYNN3.0 MAE ( m s-1)
YSU MAE ( m s-1)
Ensemble MAE( m s-1)
Point
1.37
1.29
1.18
3x3
1.36
1.28
1.17
5x5
1.36
1.25
1.16
11x11
1.38
1.18
1.14
17x17
1.39
1.16
1.13
21x21
1.40
1.17
1.14
2
Postprocessing via neighborhood approach takes a lot of computer
power, but is not improving the forecast very much
1
2
3
Table 8. MAE associated with different bias corrections developed for each PBL scheme for the
00 UTC GFS ILBC. This case study was done from 11 October 2008 to 11 November 2008.
Bias
Corrections
MYJ
(m s-1)
MYNN
2.5
(m s-1)
MYNN
3.0
(m s-1)
Pleim
(m s-1)
QNSE
(m s-1)
YSU
(m s-1)
Ensemble
(m s-1)
No Bias
2.34
2.49
2.41
2.36
2.45
2.28
2.27
Diurnal Cycle
2.29
2.33
2.28
2.27
2.30
2.21
2.18
Wind
Direction
2.27
2.27
2.26
2.29
2.28
2.24
2.17
Wind Speed
and Direction
2.15
2.16
2.14
2.17
2.17
2.10
2.05
Wind Speed
2.05
2.04
2.01
2.09
2.07
1.99
1.97
Best
Improvement
0.29 m s-1
–
Wind
Speed
0.45 m s-1
–
Wind
Speed
0.40 m s-1
–
Wind
Speed
0.27 m s-1
–
Wind
Speed
0.38 m s-1
–
Wind
Speed
0.29 m s-1
–
Wind
Speed
0.30 m s-1
–
Wind
Speed
% of
Improvement
14.1%
22.1%
20.0%
13.0%
18.4%
14.6%
15.2%
4
1
1
2
Table 10. Parameterization combinations used in the final OP ensemble to forecast wind speed
at 80 m.
Land Layer
Scheme
Initial
Boundary
Conditions
Pleim-Xiu
Pleim-Xiu
GFS
18 UTC
Pleim-Xiu
Pleim-Xiu
NAM
ACM2
00 UTC
Pleim-Xiu
Pleim-Xiu
GFS
4
YSU
00 UTC
Noah
MoninObukhov
NAM
5
YSU
00 UTC
Noah
MoninObukhov
GFS
Noah
Janjic Eta
MoninObukhov
GFS
Member
Number
PBL
Scheme
1
ACM2
18 UTC
2
ACM2
3
6
3
MYJ
Time
Land Surface
Initialization
Scheme
00 UTC
1
2
3
4
5
6
7
8
9
10
11
Table 11. MAE of final OP ensemble after wind speed bias correction compared to other six
member ensembles tested for 25 cases during the summer and fall of 2010. The deterministic
forecast is the best individual model found from the period studied. Standard devia tion (measure
of model spread) for each ensemble is also calculated. The bold value indicates a high level of
statistical improvement from the non-bias corrected six member ensembles/deterministic forecast
to the non-bias corrected final OP ensemble, with p-values less than 0.1 determined from a
Wilcoxon signed -rank test . The italics value indicates a high level of statistical improvement
from the non-bias corrected six member ensembles/deterministic forecast to the bias corrected
final OP ensemble, with p-values less than 0.1 determined from a Wilcoxon signed-rank test.
Ensemble
MAE after Bias
Correction ( m s-1)
MAE Prior to Bias
Correction ( m s-1)
Standard Deviation
after Correction ( m s-1)
GFS 00 UTC
1.67
1.99
0.74
GFS 18 UTC
1.66
2.05
0.80
NAM 00 UTC
1.68
1.91
0.67
NAM 18 UTC
1.70
1.93
0.73
Deterministic
Forecast
1.70
1.77
---
Final OP
Ensemble
1.52
1.67
0.98
1
2
3
4
5
6
7
Table 12. Number of ramp events during Day 1 (06-30 hours after model start up) and Day 2
(30-54 hours after model start up) and model error associated with ramp events for each PBL
scheme. Probability of Detection (POD), False Alarm Rate (FAR) and Threat Score were
calculated. The Bias and MAE show the timing error associated with each PBL scheme. A hit
means the model correctly predicted the ramp event within +/- 6 hours. Bold values indicate best
POD, FAR, and TS scores while underlined values indicate worst POD, FAR, and TS scores.
PBL
Scheme
Ramp
Type
Obs.
Total
Events
Model
Total
Events
Hits
False
Alarm
Miss
MAE
(hr)
Bias
(hr)
POD
FAR
TS
MYJ
Up (Day 1)
35
23
17
6
18
3.47
-1.24
0.49
0.26
0.41
Up (Day 2)
37
17
13
4
24
1.85
-1.23
0.35
0.32
0.32
Down (Day 1)
31
20
8
12
23
1.88
0.63
0.26
0.60
0.19
Down (Day 2)
35
19
12
7
23
1.42
-0.42
0.34
0.37
0.29
Up (Day 1)
35
29
19
10
16
2.68
-1.74
0.54
0.34
0.42
Up (Day 2)
37
25
15
10
22
2.33
-1.20
0.41
0.40
0.32
Down (Day 1)
31
28
11
17
20
1.64
-0.73
0.35
0.61
0.23
Down (Day 2)
35
22
11
11
24
1.55
-0.27
0.31
0.50
0.24
Up (Day 1)
35
27
17
10
18
2.88
-1.71
0.49
0.37
0.38
Up (Day 2)
37
24
16
8
21
2.75
-1.13
0.43
0.33
0.36
Down (Day 1)
31
21
9
12
22
1.89
-0.56
0.29
0.57
0.21
Down (Day 2)
35
16
8
8
27
1.50
0.25
0.23
0.50
0.19
Up (Day 1)
35
19
10
9
25
3.10
-1.30
0.29
0.47
0.23
Up (Day 2)
37
17
12
5
25
2.33
-1.83
0.32
0.29
0.29
Down (Day 1)
31
14
9
5
22
2.22
0.44
0.29
0.36
0.25
Down (Day 2)
35
20
12
8
23
2.00
0.50
0.34
0.40
0.28
Up (Day 1)
35
26
18
8
17
3.56
-2.56
0.51
0.31
0.42
Up (Day 2)
37
26
15
11
22
1.73
-1.20
0.41
0.42
0.31
Down (Day 1)
31
28
11
17
20
1.27
-1.00
0.35
0.61
0.23
Down (Day 2)
35
23
12
11
23
1.33
-0.22
0.34
0.48
0.26
Up (Day 1)
35
16
8
8
27
3.25
-0.25
0.23
0.50
0.19
Up (Day 2)
37
11
8
3
29
2.50
0.25
0.22
0.27
0.20
Down (Day 1)
31
13
9
4
22
1.33
-0.22
0.29
0.31
0.26
Down (Day 2)
35
11
9
2
26
1.33
-0.89
0.26
0.18
0.24
MYNN 2.5
MYNN 3.0
Pleim
QNSE
YSU
Persistence
Ups:
POD .38
FAR .61
TS .24
Downs:
POD .33
FAR .61
TS .21
Summary
• Multiple approaches can be used to
predict wind
• NWP with finite differencing of NavierStokes equations dominates
meteorology
• Surface friction plays major role on what
winds will be like at turbine height 
many challenges in this region
• Multiple ways to parameterize this (PBL
schemes) but all have shortcomings
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