Physics of the Atmosphere Physik der Atmosphäre. SS 2010 Ulrich Platt Institut f. Umweltphysik R. 424 Ulrich.Platt@iup.uni-heidelberg.de. Last Week. Simple energy balance calculations reveal a lot about our climate: presence of natural greenhouse effect
Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.
Institut f. Umweltphysik
Real circulation on a rotating Earth:
Three convection cells between Equator and Pole
Hypothetical circulation on a non-rotating Earth:
Just one convection cell between Equator and Pole
The term „Air Mass“ denotes an extended volume of air with unique properties (e.g. temperature, humidity, PV, vertical stability).
Definition of an Air-Mass:
- horizontal extent > 500 km
- vertical extent > 1 km
- horizontal temperature gradient < 1K / 100km
Air Masses are generated if constant conditions prevail for sufficiently long times. Prerequisites are small pressure gradients and thus only slow motion of the air mass.
Owing to these conditions, homogeneous air masses predominantly form in the tropics and in polar regions. In mid-latitudes temperature- and pressure gradients are usuallytoo strong to allow formation of distinct air masses.
Origin, Path into Europa
Island, northern ocean, northern Europe
Sub Polar air
Iceland-Greenland, North-east and Eastern Europe
Moderate Zone air
Mid Atlantic, Central- and Eastern Europe
Azors, Mediterranean, Sout-Eastern Europe
TCharacterisation of Air Masses
According to the airmass criteria X denotes not an air mass originating from mid-latitudes, but rather an air mass modified (heated, humidified) at mid-latitudes. According to their origin over land or sea, the air mass designators will be added an index c or m, respectively.
In the mid-latitude region (sub-) Tropical air borders to (sub-) polar air, thus there are large horizontal temperature gradients, which can lead to strong therma winds.
a Summerb Winter
From Malberg, 1994
The subject of hydrodynamics is the investigation and description of the motion of fluids, (i.e. liquids and gases).
Leonhard Euler, 1707-1783
Joseph Louis Lagrange,1736-1813
Point mechanics Continuum mechanics
small, but finit volume
Point mass zero extension
Temporal evolution of velocity at fixed points in space
c=„concentration“ of some quantity
The Navier-Stokes-Equations (Euler Equ.) describe the motion of a volume of fluid under the influence of the various (volume)-forces acting on it.
1) The Inertial Force
The velocity of a fluid element can change for two reasons:
1) The velocity of the entire flow changes.
2) the element is transported to a different location in the velocity field where the velocity is different.
In case 2. we obtain for e.g. the x-component:
In 3 dimensions:
The Lagrangian derivative or Material derivative describes the acceleration of a fluid element due to change of position in the flow field, it is a consequence of the Eulerian standpoint.
The expression should be interpreted as the component-wise application
of the scalar operator on , this expression is known as ‘vector gradient’.
The Material Derivative contains a non-linear term (proportional to v v), it is the dominant reason for the difficulties in solving the Navier – Stokes equations and for the unstable solutions that can arise.
Usually it is expressed in terms of the Geopotential:
Thus the gravitational force density becomes:
Vector of angular velocity in the local system:
Maximum value of FZ at Equator:
W2r = 0.034 m s-2, W2r/g ≈ 0.0035 << 1
Þ is taken up in g (or F)
Coriolis-parameter f = 2 sin , = geographical latitude
f > 0 on the northern hemisphere
f < 0 on the southern hemisphere, i.e. points into earth
In the local coordinate system we have:
vz = 0, x = y = 0
The flow is parallel to the earth’s surface vz = 0
In practice only the component z = sin of perpendicular to the surface is of importance.
With the “Coriolis Parameter” f = 2 sin = 2 z
the Coriolis force becomes:
viscosity5. The Friction force
„Shear stress-gradient force"
Shear stress, due to velocity gradients in fluids leads to friction.
Shear stress in the xy-plane xz is e.g. caused by velocity gradient of vx in z-direction
with xz = shear stress.
Note: has the dimension of a pressure (force per unit area, e.g. N/m2) but in contrast to pressure the Force vector lies in the surface, thus .
xFriction Force cont.
In a homogenous shearing stress field (linear velocity gradient) the forces on a volume element cancle. Only gradients in shearing stress give rise to a net force on a fluid volume element.
Resulting force in e.g. x-direction due to gradient in xz in z-direction:
Summing all terms we obtain the Navier-Stokes equation:
Note: here the Navier-Stokes Eq. Is given in terms of force densities, sometimes accelerations are given:
Very rough assumptions of the atmospheric spatial scales, etc.:
Horizontal/vertical scale Lh 106 m Lv 104 m Horizontal/vertical velocity vx vy 10 m/s vz 0.1 m/s Horizontal/vertical pressure gradient (p)x 10-3 Pa/m (p)z 10 Pa/m Air density 1 kg/m3 Coriolis parameter f 10-4s-1
Lagrangian derivative force:
Horizontal pressure gradient force
Vertical pressure gradient force
(molecular) Friction force
See Pressure altitdue relationship
Re > Rec ~ 1000: turbulent flow,
Flow in the atmosphere is usually turbulent
v 1 m/s, L 1000 m, n10-5 m2/s ( 0.1 cm2/s)
Re 108 >> Rec
Osborne Reynolds, 1842-1912
1) Acceleration of an air parcel by a horizontal pressure gradient.
We neglect friction force, coriolis force, gravity and Lagrange acceleration:
Force on a volume element dV:
Example: Horizontal pressure gradient:
With the air density (at sea level) 1.29 Kg/m3 we obtain the acceleration:
This is equivalent to about 14 m/s per hour. In other words the typical wind speed in the atmosphere would be reached within about one hour.
10102) The Life time of high (low) pressure systems (simplifications as in example 1):
Question: How long will it take an air-mass to cover a distance equivalent to the ‘Radius’ R of a high pressure system (see weather map)?
Covered distance s at constant acceleration a:
Witha from Example 1:
R = 500 Km
dp/dx = 510-3 N/m3 (see example 1)
corresponding to a 410-3 m/s2
In about 4 hours the air would move from the centre of the high pressure system to the rim.
In practice the lifetime of the system would be even shorter, since the ‚overpressure’ in the high pressure system would already be released, when the area A of the system had increased by:
The corresponding change in radius of the high pressure system would be:
i.e. about 25 Km
Since t s1/2 this would correspond to about 1 hour lifetime.
However high pressure systems tend to live days or weeks!?
Note: For low pressure systems entirely analogous considerations hold, just the direction of the pressure gradient (and thus the direction of the geostrophic flow) has to be reversed.
We consider a stationary flow (settling time from example 1) i.e. dv/dt=0:
x – Component of v:
With y = 0 (only the horizontal component of the coriolis force and thus z are considered) and f = 2 sin (Coriolis Parameter) we obtain:
y – Component of v:
With x = 0 we obtain:
is perpendicular to the horizontal pressure gradient.
The friction free flow under the influence of pressure gradient force and Coriolis force is called Geostrophic Wind.
The vector of the resulting wind
We neglect friction force, gravity, pressure gradient forces, and Lagrangian derivative:
Thus we obtain the radius of an inertial circle as:
At an wind speed of v0 = 5 m/s the inertial circle radii become:
= 90o: 35 Km
= 45o: 45 Km
= 15o: 130 Km
= 0o: infinite (at the Equator)
The time for one rotation is:
Example: 12 hours for = 90o
A5. The Ekman – Spiral
Close to the surface the influence of friction reduces the wind speed to levels well below the geostrophic speed vg. Since (Fc v) the influence of the Coriolis force is reduced.
Since the direction of the friction force is opposite to the direction of the wind the, close to the ground the wind direction will turn into the direction of the pressure gradient.
A) Close to the ground the friction force is relatively large, v points approximately in the direction of pressure gradient force.
B) In intermediate altitudes there is already a considerable angle between FP and v.
C) In the geostrophic case (at several 100 m altitude) the friction force can be neglected and FC is anti parallel to FP. The air parcel moves at right angle to the pressure gradient force.
Schematic view of the orientation of the wind vector at different altitudes (Zg 1500 m)
[Guyot, Physics of the Environment and Climate, WILEY, 1998]
Wind direction boundary layer
"Leakage" only close to the ground because of the Ekman - Spiral
1010 hPa6. Life time of a high pressure system including Ekman – spiral, or: Why is there fair weather under high pressure?
High pressure system, top view
The descending air is heated due to adiabatic compression
Usually clouds dissolve
High pressure system, side view
An air-parcel (volume: V) with a potential temperature which deviates by from the temperature of the surrounding air will deviate in density from its surroundings:
and thus experience the lifting force:
F per unit volume:
and the vertical acceleration:
= +1 K, = 300 K az = 3.310-2 ms-2
The air-parcel will cover the distance from the earth’s surface to the tropopause (assumed at 10 Km altitude) in the time t:
In components, with
Note: Anisotropy between horizontal and vertical flow, because of gravity and different lenght scales
Þ frequently: v1 ~ v2 ~ vh >> v3 = vz
With f = 2W·sinj (=Coriolis parameter):
O-W3) Geostrophic Flow
Geostrophic flow: Stationary flow without friction
From the equilibrium pressure gradienten force = Coriolis force we obtain the geostrophic flow velocities:
Geostrophic flows: perpendicular to the pressure gradient.
They do not reduce pressure differences!
Rule of thumb for the direction of geostrophic flows:
N-Hemisphere: Higher pressure to the right of the flow
S-Hemisphere: The other way around
= Geographical latitude
Frontal zones form between two air masses of different temperature. In a narrow inner region temperature gradients can become very large (up to 5K/100km); this region is called a front. At a given altitude the horizontal ‘extent’ of a front is several 10 km. Since fronts are tilted (as will be detailed), their total horizontal extent can reach several 100 km.
Within a front the isotherms are strongly tilted; there is a pronounced baroclinic structure. This is frequently called a hyperbaroclinic structure.
Air masses, frontal zone, front al region, and front. Malberg, 1994
We assume a front oriented along the x-direction(being infinitely extended in x-direction).
T = 10 K, v = 50 m/s
For 50° geographical latitude (f = 11.2 10-5 s-1) follows:
Thus the frontal zone will extend horizontally over 8 km/0.171 ≈ 470 km at 8000 m vertical extent.
In reality there are also wind components perpendicular to the front, thus the front will move. In addition, friction at the surface will then play a role.
Cold Front Warm Front
Typical structures of cold fronts and warm fronts
Right: Rising heated air (usually in a rather confined area, the „hot tower“). Due to its high potential temperature it can reach up to the stratosphere.
Left: Connection to the mean circulation of higher latitudes.
Centre: large-area subsiding air of the trade wind zone.
Hatched area: wind maximum of the subtropical jet (STJ).
Vertical scale exaggerated.
Schematic cross-section through the Hadley-cell.
We consider an airmass with the angular momentum L, and momentum of inertia J, both quantities per unit mass and with respect to the Earth‘s axis.
With v: zonal wind speed (relative to Earth): geographical latitudeR: radius of Earth:
after Lamb 1972 and Gross 1972, from Roedel 2000
Meridional (E-W), zonal (N-S), and interhemispheric mixing times of the atmosphere (D.J. Jacob 1999)
Maiss and Levin 1994, GRL 21(7), 569-572
courtesy Levin & Engel