Basics of Wind Meteorology - Dynamics of Horizontal Flow

WIND ENERGY METEOROLOGY
SS 2011

Detlev Heinemann

ENERGY METEOROLOGY GROUP
INSTITUTE OF PHYSICS
OLDENBURG UNIVERSITY
FORWIND – CENTER FOR WIND ENERGY RESEARCH

Montag, 18. April 2011


Contents

I. Basic Meteorology
- Dynamics of Horizontal Flow
(forces, equation of motion, geostrophic wind,
frictional effects, primitive equations, general circulation)
- Atmospheric Boundary Layer
(turbulence, vertical structure, special BL effects)
II. Atmospheric Flow Modeling
- Model classes: Linear, RANS, LES, ..
- Application: Wind farm modeling
III. Offshore-Specific Conditions
IV. Resource Assessment & Wind Power Forecasting
V. Wind Measurements & Statistics

2


INTRODUCTION

Research related to wind energy
meteorology at the Institute of Physics /
ForWind

3


RESEARCH TOPICS

Forecasting of wind power
Offshore wind energy meteorology
Numerical modelling of wind flow
Turbulent characteristics of wind flow

4


WIND POWER FORECASTING

Numerical Weather Prediction
Wind speed, direction

Spatial Refinement
roughness, orography, thermal stability Forecasting
local
Wind farm power output
power characteristics, shading losses

Correction of systematic errors

28
13
5



PHYSICAL MODELS
Input:
- wind speed forecasts at hub height
- roughness parameter / orography
- thermal stability
- wind farm geometry
- power curve
- produced power ('measurement')
for correction (model output statistics)

6



STATISTICAL MODELS

Input:
- wind speed forecasts at hub height
- produced power ('measurement')
as training data (e.g., in a Neural Net)

statistically derived wind power curve includes:
- wind farm effects (wake effects)
- regional/local situation (roughness, orography, etc.)
- regular updates ensure adaptation to changes

7


Example: Vertical wind profile

comparison of different
theoretical vertical profiles
with IEC standard

large deviations of real
profiles
importance of
atmospheric stability over
the ocean

8


Offshore Wind Energy

9


Offshore Wind Energy

measurement
platform FINO-1
test field Alpha
Ventus

9


Numerical Modelling of Atmospheric Boundary
Layer Flow
Application: Offshore, complex terrain, thermally induced flow

Tasks: - Parametrisation of local and small-scale effects (turbulence!)
- coupling of differnet scales
e.g: meso scale models and Large Eddy Simulation (LES)
to couple the synoptic scale flow and wakes behind
wind turbines

important for: turbine design, resource assessment, forecasting

10


Offshore Wind Resources

Modelling the atmospheric boundary layer wind field
in offshore wind farms

extension of knowledge of the marine atmopheric boundary layer
with respect to wind energy applications
vertical structure of wind fields over the ocean
turbulence in offshore wind farms
influence of wakes in large offshore wind farms on local wind fields
modlling the influence of air sea interaction
interaction of wind and waves
reliable data for turbine design

11


OFFSHORE WIND RESOURCES

Meso Scale Modelling

high resolution up to 1 km
Resource assesment and forecasting
Offshore and coastal regions
complex terrain
extreme events on long time scales
time scales from hours to decades
no small-scale turbulence resolved

12


OFFSHORE WIND RESOURCES

mean wind speed
in m/s for the period
2004-2006.

calculated with the
mesoscale model WRF
and data from the
measurement platform
FINO-1

13


WAKE MODELLING IN WIND FARMS

Calculation of wind speed
deficit in single wake with
Ainslie model (Reynolds-Solver)
‣Superposition of multiple
wakes (wind farm situation)
‣Influence of turbulence
intensity on wake shape
‣Estimation of yearly power
production based on the wind
speed distribution
‣Application of Large Eddy
Simulation (LES)

14


TURBINE DYNAMICS

wind induced turbine
dynamics are in time scale
of sec and below

knowledge of wind
characteristics in time
scale of sec necessary

15

Windböe - was ist dies?

WIND GUSTS
uτ := v(t + τ ) − v(t) ur := v(x + r) − v(x)

16


STATISTICS OF WIND GUSTS
(Wind fluctuations)

P(uτσ−1)
τ=4s

1/hour

~106

1/100 years

Boundary-Layer Meteorology 108 (2003)
17


WIND TURBINE POWER CURVES:
data sheets vs. reality

Wind turbine power output is result of nonlinear dynamic processes

But:
power curve P(v) is usually taken
from simplified data sheets

18


WIND TURBINE POWER CURVES:
data sheets vs. reality

individual power curves according to the meteorological
situation

 governing parameter:
- wind direction,
- atmospheric stability,
- turbulence intensity
 aim: „learning“ power curves
 integration in forecasting schemes

19


I BASIC METEOROLOGY

I-1 Dynamics of Horizontal Flow

20


Dynamics of Horizontal Flow
Newton’s second law

in each of the three directions in the If the coordinate system is accelerated,
coordinate system, the acceleration a apparent forces are introduced to
experienced by a body of mass m in compensate for this acceleration of the
response to a resultant force ΣF is given coordinate system.
by
In a rotating frame of reference two
different apparent forces are required:
‣ a centrifugal force that is experienced
by all bodies, irrespective of their
motion,
This equation describes the motion in an ‣ and a Coriolis force that depends on
inertial (i.e. nonaccelerating) frame of the relative velocity of the body in the
reference. plane perpendicular to the axis of
rotation (i.e., in the plane parallel to
the equatorial plane).

21


Real Forces

‣ Gravitation
‣ Pressure gradient force
‣ Frictional force

22


Insert: Total & local time derivatives
Atmospheric variables typically depend on both time and space:
ψ = ψ(t,x,y,z)

total time derivative d/dt
rate of change following an air parcel as it moves along its three-
dimensional trajectory through the atmosphere (Eulerian)

local derivative ∂/∂t
rate of change at a fixed point in rotating (x, y, z) space
(Lagrangian)

Related by chain rule:

advection terms

23


Hydrostatic Equation & Geopotential (I)
Atmopheric pressure at any height is
due to the force per unit area exerted
by the weight of the air above that
+ height.
--> atmospheric pressure decreases
with increasing height
Net upward force due to the
decrease in atmospheric pressure
with height: -δp
Wallace & Hobbs (2006)
Net downward force due to gravi-
tational force acting on the slab: gρδz

If the net upward force on the slab equals the downward force:
Atmosphere is in hydrostatic balance.

24


Hydrostatic Equation & Geopotential (II)
For an atmosphere in hydrostatic balance, the
balance of forces in the vertical requires that

Note: δp is negative!

or, with δz -> 0:
Balance of
gravitational force
Hydrostatic Equation and
vertical component of
pressure gradient force
Integration then yields:

25


Hydrostatic Equation & Geopotential (III)
The geopotential at any point in the Earth’s atmosphere is defined
as the work that must be done against the Earth’s gravitational
field to raise a mass of 1 kg from sea level to that point.
In other words, is the gravitational potential per unit mass.
units of geopotential: Jkg-1 or m2s2.

dΦ = gdz = - 1/ρ dp

The geopotential Φ(z) at height z is thus given by

with Φ(z=0) = 0 at sea level.

26


Hydrostatic Equation & Geopotential (IV)
Definition of the geopotential height Z:

g0 is the globally averaged acceleration due to gravity at the Earth’s
surface (9.81ms-2).
Geopotential height is often used as the vertical coordinate in
atmospheric applications in which energy plays an important role
(e.g., in large-scale atmospheric motions).
The values of z and Z are almost the same in the lower atmosphere
where g≅g0.

27


Pressure Gradient Force

The pressure gradient
force is directed down
the horizontal pressure
gradient ∇p from higher
toward lower pressure.

The x-component of the pressure gradient force
acting on a fluid element:

The horizontal components of the pressure gradient
force and acceleration, respectively, then are:

28


Frictional Force
frictional force (per unit mass):

τ represents the vertical compo- Free atmosphere (above the
nent of the shear stress (i.e., the boundary layer):
rate of vertical exchange of hori- Frictional force << pressure
zontal momentum) in units of gradient force, Coriolis force
Nm-2 due to the presence of smal- Within the boundary layer:
ler, unresolved scales of motion. Frictional force ~ other terms in
the horizontal equation of motion

29


Shear Stress
The shear stress σs at the Earth’s surface is in the opposing
direction to the surface wind vector Vs.

Approximation by the empirical relationship:

where
ρ density of the air
CD dimensionless drag coefficient (varying with
surface roughness and static stability
Vs surface wind vector
Vs (scalar) surface wind speed
30


The Coriolis Effect

© Commonwealth of Australia, Bureau of Meteorology, 2006
The Coriolis effect describes an 'apparent' force that causes 'apparent' deflections. It
increases with increasing latitude and wind speed, and alters the direction of the wind,
but not its speed.
The Coriolis force can therefore balance the pressure force so that, in the northern
hemisphere, the air will flow anticlockwise around a centre of low pressure and
clockwise around a centre of high pressure.

31


Coriolis Force – Mathematical Description
transformation of coordinates between the inertial reference frame and
1 the reference frame rotating with the angular velocity of the earth ...

2 … and applied to the wind velocity vector v=d‘r/dt …

…and substituting (1) in (2) adds two new
3 components: the Coriolis acceleration (2nd
term) plus the centripetal acceleration (3rd
term)

4 The Coriolis force and acceleration in vector notation …

5
… and the horizontal component only.

32


Coriolis Force – Properties

‣ The Coriolis force is proportional the the object’s velocity, i.e., it is only
acting on moving objects.
‣ The Coriolis force acts perpendicular to the direction of a moving object.
‣ In the northern hemisphere this results in a deflection of the horizontal
wind vector to the right, in the southern hemisphere to the left.
‣ Consequently, the Coriolis force only affects the direction, not the velocity.
No work on the object is performed.
‣ The Coriolis force vanishes at the equator and is maximum at the poles.

Ω = (0, Ω cos φ , Ω sin φ) is the vector of the earth’s rotation with (|Ω| = 7.29 · 10−5 rad s−1).
f = 2 Ω sin φ ( ~10 −4 s −1 in midlatitudes) is the Coriolis parameter.

33


Equation of Motion (I)
Horizontal motions of fluid In component form:
elements in the atmosphere are
governed by the acting forces:
Fh = Fp,h + Fc,h + Ffr,h
The individual acceleration of fluid
elements (air parcels) thus is:
dvh/dt = ah = ap,h + ac,h + afr,h 1) Notes:
1) dvh/dt is the Lagrangian time derivative
Then the horizontal equation of
motion can be written: of the horizontal velocity component
experienced by an air parcel as it moves
about in the atmosphere.
2 ) Accelaration is due to a change in
velocity of the motion as well as due to
a change in direction of the motion.

34


Equation of Motion (II)
The density dependence can be eliminated by substituting
Fp = -1/ρ ∇p by Fp = - ∇Φ:

Here, the horizontal wind field is defined on surfaces of constant
pressure (∇p=0) instead of surfaces of constant geopotential (∇Φ=0).

35


The Geostrophic Equilibrium
The geostrophic equilibrium is a state of motion of an inviscid fluid in which
the horizontal Coriolis force exactly balances the horizontal pressure force
at all points of the field:

f (k × vh) = - (1/ρ) ∇H p

where f is the Coriolis parameter, k the vetical vector of unity, vh the horizontal
wind vector, ρ the density of air, p the pressure, and ∇H the horizontal gradient
operator.
With respect to cyclone-scale motions in extratropical latitudes, the free
atmosphere frequently approaches a state of geostrophic equilibrium.

The horizontal gradient operator is:

© American Meteorological Society, Glossary of Meteorology 36


The Geostrophic Wind
The geostrophic wind is the horizontal wind velocity for which the Coriolis
acceleration exactly balances the horizontal pressure force:

f k × vg = - g ∇p z

where vg is the geostrophic wind, f the Coriolis parameter, k the vertical
unit vector, g the acceleration of gravity, ∇p the horizontal del operator with
pressure as the vertical coordinate, and z the height of the constant-
pressure surface.

The geostrophic wind is thus directed along the isobars in a geopotential
surface with low pressure to the left in the Northern Hemisphere and to the
right in the Southern Hemisphere.

The geostrophic wind is defined at every point except along the equator.

© American Meteorological Society, Glossary of Meteorology 37


The Geostrophic Wind: Example

Low pressure system over Great
Britain

Δp = 32 hPa
Δx = 600 km

latitude: Φ = 54°N
Coriolis parameter:
f = 2 Ω sinΦ = 1.18 10-4 s -1

vg = - 1 / (1.2 kgm -3 x 1.18 10-4 s -1) x
(32 hPa / 0.6 106 m)

= 38 ms-1

38


Balances of the Horizontal Wind Field

The geostrophic balance Balance of boundary layer flow
The horizontal components of the The pressure gradient force Fp,h is balanced
pressure gradient force Fp,h and the by the sum of the Coriolis force Fc,h and the
Coriolis force Fc,h are balanced. vg is the frictional force Ffr.
geostrophic wind. The stronger the frictional force Ffr, the larger
the angle between vfr and vg and the more
subgeostrophic the surface wind speed vfr.

39


The Primitive Equations (I)
The horizontal equation of motion is part of a complete system of
equations that governs the evolution of large-scale atmospheric
motions – the socalled primitive equations.

The other primitive equations relate to the vertical component of
the motion and to the time rates of change of the thermodynamic
variables p, ρ, and T.

Equations containing time derivatives are prognostic equations. The
remaining so-called diagnostic equations describe relationships
between the dependent variables that apply at any instant in time.

40


The Primitive Equations (II)

horizontal equation of motion

hydrostatic/hypsometric equation

thermodynamic energy equation
(κ=0.286, ω=dp/dt)
continuity equation

Five equations in five dependent variables: u, v, ω, Φ, and T.
The fields of diabatic heating J and friction F need to be parameterized.

41

Basics of Wind Meteorology - Dynamics of Horizontal Flow

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