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N. Kämpfer
Atmospheric
Dynamics
Atmospheric Dynamics
Introduction
Momentum
equation
N. Kämpfer
Geostrophic wind
Thermal wind
Institute of Applied Physics
University of Bern
Primitive equations
Vorticity
Outline
Introduction
N. Kämpfer
Atmospheric
Dynamics
Momentum equation
Introduction
Momentum
equation
Geostrophic wind
Geostrophic wind
Thermal wind
Thermal wind
Primitive equations
Vorticity
Primitive equations
Vorticity
Introduction
The circulation of the atmosphere of a planet is a key
component of its climate. In case of ♁ for example:
I
I
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Atmospheric motions carry heat from the tropics to the
pole
winds transport humidity from the oceans to the land
Distribution of most chemical species in the atmosphere
is the result of transport processes
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O3 is mainly produced in equatorial regions but is
transported polewards
H2 O enters the middle atmosphere in the tropics and is
moved toward the poles
CFCs are released in industrial areas and generate
ozone-holes in polar regions
Aerosols are produced in industrial areas and affect
regions far away
N. Kämpfer
Atmospheric
Dynamics
Introduction
Momentum
equation
Geostrophic wind
Thermal wind
Primitive equations
Vorticity
Introduction
Air motions are strongly constrained by:
I
I
Density stratification
→ gravitational force resists vertical displacement
Earth rotation
→ Coriolis force is a barrier against meridional
displacements
N. Kämpfer
Atmospheric
Dynamics
Introduction
Momentum
equation
Geostrophic wind
Transport occurs at a variety of spatial and temporal scales.
In case of ♁:
I global
I
synoptic ≈ 1000km
→ e.g. high and low pressure systems
I
mesoscale ≈ 10 - 1000km
→ e.g.fronts
I
small scale
→ e.g. planetary boundary layer
Thermal wind
Primitive equations
Vorticity
Introduction
In order to describe the dynamical behavior of the
atmosphere, we treat it as a fluid
→Fundamental equations of fluid mechanics must be used
N. Kämpfer
Atmospheric
Dynamics
Introduction
Circulation of a planets atmosphere is governed by three
basic principles:
Momentum
equation
Geostrophic wind
1. Newton’s law of motion
Thermal wind
2. Conservation of enetgy → 1.st law of thermodynamics
Primitive equations
3. Conservation of mass → equation of continuity
Vorticity
plus the equation of state
In using these laws we must consider that we are operating
in a rotating frame of reference!
→ we have to consider centrifugal and the Coriolis force
Coordinate system
It is practical to use a spherical coordinate system to
describe dynamical aspects
The coordinate system is rotating together with the planet
We use:
λ
φ
z
R
longitude in easterly direction
latitude in northerly direction
geometric altitude
radius of planet
N. Kämpfer
Atmospheric
Dynamics
Introduction
Momentum
equation
Geostrophic wind
Thermal wind
Primitive equations
We define:
r =R +z
distance from the center of the planet
u = r cos φdλ/dt zonal component of wind velocity ~v
measured towards the east
v = rdφ/dt
meridional component of wind velocity ~v
measured towards the north
w = dz/dt
vertical component of ~v
Vorticity
Real forces
Real forces entering in the equation of motion are:
1. Pressure gradient force, F~p
Whenever there is a gradient in pressure the resulting
force per mass is given by
1~
F~p = − ∇p
ρ
F~p is normal to isobars and is directed from higher to
lower pressure
Distance of isobars→ measure for the pressure gradient
Typical values for pressure gradient for ♁:
→ ca. 1mb per 8m in vertical direction and
1mb per 10km in horizontal direction
2. Frictional force, F~R is nearly proportional to wind speed
N. Kämpfer
Atmospheric
Dynamics
Introduction
Momentum
equation
Geostrophic wind
Thermal wind
Primitive equations
Vorticity
F~R = −a~v
important in planetary boundary layer
3. Gravity force, F~G
Apparent forces
An object moving with velocity ~v in a plane perpendicular to
the axis of rotation experiences an apparent force, called
Coriolis force, F~C .
F~C = 2~v × ω
~
The angular velocity is for ♁: ω = Ω = 7.3 · 10−5 sec −1
The Coriolis force per mass for zonal velocity, u, and
geographic latitude ϕ is
dv
= −2Ωu sin ϕ
dt
and analogue for meridional direction
du
= 2Ωv sin ϕ
dt
where f is called Coriolisparameter
f = 2Ω sin ϕ
f is equal to zero at the equator and a maximum at the
poles f = ±1.46 · 10−4 sec −1
N. Kämpfer
Atmospheric
Dynamics
Introduction
Momentum
equation
Geostrophic wind
Thermal wind
Primitive equations
Vorticity
Apparent forces
The Coriolis force in horizontal direction can be written as
F~CH = −f ~k × ~v
~k is a unit vector in z-direction
N. Kämpfer
Atmospheric
Dynamics
Introduction
The Coriolis force is acting normal to the direction of
motion
→ to the right in the northern hemisphere
→ to the left in the southern hemisphere
Momentum
equation
I
As the force is acting normal to the direction of motion
no work is performed
Vorticity
I
The force vanishes at the equator and is maximum at
the poles
I
Geostrophic wind
Thermal wind
Primitive equations
The centrifugal force is another force showing up in a
rotating reference frame
→ its effect is ”absorbed” in the gravitational force
Equation of motion
Considering all forces: general equation of motion per mass
d~v
~p + F
~c + F
~R + F
~G
=F
dt
N. Kämpfer
Atmospheric
Dynamics
resp.
d~v
1~
~ × ~v − a~v + ~g
= − ∇p
− 2Ω
dt
ρ
Care has to be taken what is meant by the time derivative!
Introduction
In the Eulerian frame this is for a fixed grid of points,
actually identical to ∂~v /∂t
Alternative: follow the flow In the Lagrangeian frame. This
leads to the so called material derivative
d
∂
~
=
+ ~v · ∇
dt
∂t
and therefore
d~v
∂~v ~ =
+ ~v · ∇ ~v
dt
∂t
→ nonlinear in ~v → difficult to forecast atmospheric state
Thermal wind
Momentum
equation
Geostrophic wind
Primitive equations
Vorticity
Geostrophic wind
Consider equation of motion in horizontal direction
d~v
dt
N. Kämpfer
~ p + F~c + F~R
= F
H
Atmospheric
Dynamics
1~
= − ∇p
− f ~k × ~v − a~v
ρ
Above approx. 1 km friction can be neglected
Air parcel starts to move due to pressure gradient → evokes
Coriolis force → deviation of track to the right → equilibrium
Introduction
Momentum
equation
Geostrophic wind
Thermal wind
Primitive equations
Vorticity
1~
f ~k × ~v = − ∇p
ρ
Vector multiplication with ~k on the left allows to solve for ~v
Velocity v~g is called geostrophic wind
v~g =
1 ~ ~
k × ∇p
ρ·f
Geostrophic wind
v~g =
1 ~ ~
k × ∇p
ρ·f
fv =
1 ∂p
ρ ∂x
where
f = 2Ω sin ϕ
− fu =
1 ∂p
ρ ∂y
Note:
N. Kämpfer
Atmospheric
Dynamics
Introduction
Momentum
equation
I
Geostrophic wind is parallel to isobars
Geostrophic wind
I
On northern hemisphere → low pressure system on the
left
Thermal wind
Primitive equations
Vorticity
I
Wind around low pressure system in the same direction
as Earth rotation
I
The denser the isobars the higher the wind speed
I
As geostrophic wind is parallel to isobars (normal to
gradient) → pressure imbalance can not be changed
I
When friction is present → subgeostrophic wind →
pressure imbalance can be changed
Thermal wind
From geostrophic wind equation together with ideal gas law
N. Kämpfer
1 ∂p
RT ∂p
∂ ln p
fv =
=
= RT
ρ ∂x
p ∂x
∂x
Atmospheric
Dynamics
Introduction
From hydrostatic balance
−
Momentum
equation
∂ ln p
g
=
RT
∂z
Geostrophic wind
Thermal wind
Primitive equations
Neglecting vertical variations in T we get
f
g ∂T
∂v
≈
∂z
T ∂x
f
Vorticity
∂u
g ∂T
≈−
∂z
T ∂y
These are the thermal wind equations
They give relations between horizontal temperature
gradients and vertical gradients of the horizontal wind when
both geostrophic and hydrostatic balance apply
Thermal wind
N. Kämpfer
Atmospheric
Dynamics
Introduction
Geostrophic wind
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Momentum
equation
tl
(r5
\\
Primitive equations
NJ
Vorticity
\\
trv
l
I
hr
vv
Thermal wind
from Dutton: Dynamics of atmospheric motion
Latitudinal temperature gradient causes an increase in the
latitudinal pressure gradient → geostrophic wind speed also
increases with height
Thermal wind
N. Kämpfer
Example: The zonally average zonal wind in the lower and
middle atmosphere of the Earth is close to a thermal wind
Atmospheric
Dynamics
Introduction
Momentum
equation
Geostrophic wind
Thermal wind
Primitive equations
Vorticity
from Brasseur: Aeronomy of middle atmosphere
from Brasseur: Aeronomy of middle atmosphere
Primitive equations
The whole set of equations used to describe atmospheric
dynamics is called primitive equations
d~v
1~
~ × ~v − a~v + ~g
= − ∇p
− 2Ω
dt
ρ
N. Kämpfer
Atmospheric
Dynamics
Introduction
Momentum
equation
the gas law
p = ρRT
Geostrophic wind
Thermal wind
the first law of thermodynamics
Primitive equations
Vorticity
R T dp
Q̇
dT
=
+
dt
cp ρ dt
cp
and the continuity equation
dp
~ · ~v = 0
+ ρ∇
dlt
Primitive equations
The whole set in spherical coordinates
du
uv
1 ∂φ
=
tan ϕ + 2Ω sin ϕv −
+ Fλ
dt
R
R cos ϕ ∂λ
dv
u2
1 ∂φ
= − tan ϕ − 2Ω sin ϕu −
+ Fϕ
dt
R
R ∂ϕ
∂φ
RT
=
∂z
H
dT
RT
Q̇
+w
=
dt
cp H
cp
N. Kämpfer
Atmospheric
Dynamics
Introduction
Momentum
equation
Geostrophic wind
Thermal wind
Primitive equations
Vorticity
1
∂u
1 ∂(v cos ϕ) 1 ∂ρ · w
+
+
=0
R cos ϕ ∂λ R cos ϕ
∂ϕ
ρ ∂z
d
∂
u
∂
v ∂
∂
=
+
+
+w
dt
∂t
R cos ϕ ∂λ R ∂ϕ
∂z
where Φ = gz is the geopotential and F are friction force
components
Vorticity
In addition to the primitive equations also equations
describing vorticty in a fluid field are of importance
Vorticity, ζ, in a horizontal flow is the vertical component of
the rotation of the velocity field
~ z × ~v = ∂vy − ∂vx = ∂v − ∂u
ζ=∇
∂x
∂y
∂x
∂y
N. Kämpfer
Atmospheric
Dynamics
Introduction
Momentum
equation
Geostrophic wind
Circulation is defined as
Thermal wind
I
Z=
~
~v · ds
According Stokes:
~ n × ~v = d
∇
dA
I
Vorticity shows up in two cases:
I
I
in flows that are bending
in straight motion with shear
~
~v · ds
Primitive equations
Vorticity
Vorticity
Simplest case: v = ω · r → Circulation is then
I
~ = v · 2πr = 2πr 2 ω
Z = ~v · ds
and vorticity by division of the area element
N. Kämpfer
Atmospheric
Dynamics
Introduction
Momentum
equation
Z
ζ = 2 =2·ω
r π
Geostrophic wind
In this case vorticity is just two times the angular velocity
In general
dZ
v (r ) ∂v
ζ=
=
+
dA
r
∂r
So far we have discussed the local vorticity, ζ. In an inertial
reference system we have to add the part stemming from the
planetary rotation, i.e. ω = 2Ω sin ϕ = f
The absolute vorticity, η,is thus
Thermal wind
Primitive equations
Vorticity
η =ζ +f
Vorticity
For a geostrophic wind, we have
~vg =
1~ ~
k × ∇p
ρf
N. Kämpfer
Atmospheric
Dynamics
and therefore
Introduction
~ z × ~vg + f = ∇
~z ×
η =ζ +f =∇
1~ ~
k × ∇p + f
ρf
Momentum
equation
Geostrophic wind
Thermal wind
η=
1 2
∇ p+f
ρf
It can be shown that total vorticity is concerved
dη
~ H · ~v = 0
+ η∇
dt
where
∂η
dη
~ Hη
=
+ ~v · ∇
dt
∂t
Primitive equations
Vorticity
Vorticity
Air parcels deflected across latitudinal circles adjust to
changes in local rotation → preserve absolute angular
momentum as seen in an inertial frame
A measure of this is potential vorticity defined as
f +ζ
PV =
h
N. Kämpfer
Atmospheric
Dynamics
Introduction
Momentum
equation
Geostrophic wind
Thermal wind
To leading order, absolute
vorticity η = ζ + f is
constant: the relative
vorticity ζ simply being
exchanged with the
planetary vorticity f
→ Planetary waves set up
Primitive equations
Vorticity
Hurricane on Venus
N. Kämpfer
Atmospheric
Dynamics
Introduction
Momentum
equation
Geostrophic wind
Thermal wind
Primitive equations
Vorticity