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### 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
- latitudinal and vertical structure of T and energy
- radiative-convective equilibrium
- Feedbacks are important and can only be addressed with numerical models
- The global energy budget is in delicate balance, small changes have large effects

Outline for Today

- Intro. to Atmospheric Dynamics
- Global Circulation
- The Intertropical Convergence Zone
- Origin of High- and low pressure systems
- Interhemispheric exchange

Global Tropospheric Circulation

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 Concept of Air Masses

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.

Designation

Origin, Path into Europa

Polar Air

T

Island, northern ocean, northern Europe

Sub Polar air

PS

Iceland-Greenland, North-east and Eastern Europe

Moderate Zone air

X

Mid Atlantic, Central- and Eastern Europe

Subtropical Air

TS

Azors, Mediterranean, Sout-Eastern Europe

Tropical Air

T

Characterisation of Air MassesAccording 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.

The subject of hydrodynamics is the investigation and description of the motion of fluids, (i.e. liquids and gases).

- We distinguish between two fundamentally different types of representation:
- 1.) Description of flow as the temporal development of the local velocity field in a given point in space
- Eulerian Representation
- 2.) Description of flow by the temporal development of the position of a particular element of mass of the fluid in space.
- Lagrangian Representation
- In the Lagrangian Representation the velocity is then obtained from the temporal derivative:

Leonhard Euler, 1707-1783

Joseph Louis Lagrange,1736-1813

Point Mechanics vs. Continuum Mechanics

Point mechanics Continuum mechanics

small, but finit volume

Point mass zero extension

Lagrangian representation:

Eulerianrepresentation:

Temporal evolution of velocity at fixed points in space

?

Conservative Quantities in Space and Time

c=„concentration“ of some quantity

The Navier – Stokes – Equations (Conservation of Momentum)

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 (Material Derivative)

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.

Comment 1:

The expression should be interpreted as the component-wise application

of the scalar operator on , this expression is known as ‘vector gradient’.

Comment 2:

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.

F(x)

F(x+x)

A

x

The Pressure Gradient ForceForce on the left and right surface, respectively, of a (cubic) fluid volume:

Net force on the fluid volume:

Force per volume:

In three dimensions:

The Gravity Force

Usually it is expressed in terms of the Geopotential:

Thus the gravitational force density becomes:

Inertial Forces due to Earth‘s Rotation

Vector of angular velocity in the local system:

Zentrifugal force:

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)

4. The Coriolis force

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:

The Coriolis Force (2)

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:

vx

dynamic

kinematic

viscosity

5. 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 .

vx(z)

z

FRx(z+dz)

(z+dz)

z+dz

(z)

y

z

FRx(z)

x

Friction 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:

In 3-D:

The Navier-Stokes Equation

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:

Magnitude of Terms in the Navier-Stokes Equation

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

Gravitational force

Coriolis force

(molecular) Friction force

See Pressure altitdue relationship

The Reynolds Number:

- Comparison of terms of the Navier-Stokes eq. (forces or accelerations) yields dimensionless „numbers“, e.g.:

- Reynold number:

Re > Rec ~ 1000: turbulent flow,

Flow in the atmosphere is usually turbulent

Example:

v 1 m/s, L 1000 m, n10-5 m2/s ( 0.1 cm2/s)

Re 108 >> Rec

Osborne Reynolds, 1842-1912

Examples for the Application of the Navier-Stokes Equation:

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:

We retain:

Its acceleration:

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.

R

1020

1010

2) 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:

Numerical example:

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.

3. Friction free motion of an air-parcel under the Influence of Coriolis and Pressure Gradient Force

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:

or

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.

or

The vector of the resulting wind

4. Purely inertial motion of an air-parcel

We neglect friction force, gravity, pressure gradient forces, and Lagrangian derivative:

Centrifugal force:

Thus we obtain the radius of an inertial circle as:

Numerical Example:

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

C

A

5. The Ekman – SpiralClose 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.

The Ekman – Spiral (2)

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 free atmosphere

Wind direction boundary layer

Descending air

"Leakage" only close to the ground because of the Ekman - Spiral

H

R

1020 hPa

1010 hPa

6. 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

fair weather

High pressure system, side view

7. Lifting of an air-parcel

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:

Numerical example:

= +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:

1) Navier-Stokes Eq. in Components

In components, with

Note: Anisotropy between horizontal and vertical flow, because of gravity and different lenght scales

Þ frequently: v1 ~ v2 ~ vh >> v3 = vz

1020

1000

East-West

North-South

N-S

O-W

3) Geostrophic FlowGeostrophic 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

4) Definition of Vorticity

- The curl of the wind vector field is only important in the horizontal since the vertical extent of the atmosphere is very small
- The relative vorticity (relative vortex strength) is defined as the z-component of the curl of the wind vector field:with:

5) Example: Vorticity of a Rigid Rotator

- Velocity at distance r from the axis:
- „Circulation“:
- Thus, the vorticity is
- The vorticity of a rigid rotator is twice its angular velocity
- Example: High pressure system, R = 500 km, v=10 ms-1 = v/r ≈ 10/5·105 = 2·10-5 s-1 = 2 = 4·10-5 s-1

6) The Absolute Vorticity

- The (large scale) dynamics of fluids on Earth is described in a rotating coordinate system
- The Earth is a rigid rotator with angular velocity Ω
- Local vorticity in z-direction at latitude φ is given by the Coriolis parameter f = 2 Ω sin φ(see previously discussed vorticity of a rigid rotator)
- Thus the absolute vorticity η of the wind field is the sum of the relative vorticity ζ (measured relative to the terrestrial coordinate system) and the Coriolis parameter f:

= Geographical latitude

Frontal Zones

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

Tilt of Fronts – Margules‘ Formula

- The tilt of a front is determined by two counteracting forces:
- - Due to ist lower density the warmer air mass has the tendency to move above the colder air mass.
- The Coriolis force generated by the dominating wind direction parallel to the front acts against the former force.
- Taking into account that at either side of a stable front there has to be equal pressure. Thus, the change of pressure along an infinitesimal distance ds (see Figure) on the warm and cold side has to be equal:

We assume a front oriented along the x-direction(being infinitely extended in x-direction).

z

warm

v

k

ds

cold

y

a

x

ds

dz

dy

Tilt of Fronts – Margules‘ Formula (1)Taking into account that at either side of a stable front there has to be equal pressure. Thus, the change of pressure along an infinitesimal distance ds (see Figure) on the warm and cold side has to be equal:

Margules‘ Formula (4)

Example:

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.

Fronts in Motion

Cold Front Warm Front

Mean Flow

Roedel 2000

Typical structures of cold fronts and warm fronts

Air Masses, Cold Front, and Jet Stream during a Storm on Nov. 12, 1972

Cyclogenesis

Roedel 2000

The Hadley Cell and Inter-Tropical Convergence Zone

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.

From: Roedel

Schematic cross-section through the Hadley-cell.

Roedel 2000

The Origin of the Subtropical Jet (STJ)

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:

18

tropopause

14

= 330 K

10

release of latent heat

Altitude (km)

Tropical low pressure

(convergence)

Subtropical high

Pressure (divergence)

The Hadley CirculationPosition of the ITCZ and Mean Wind Directions

ITCZ

NH Summer

Equator

Tropic

NH Winter

after Lamb 1972 and Gross 1972, from Roedel 2000

Position of the ITCZ and Mean Pressure

Roedel 2000

The Circulation Pattern of the Hadley Cell

Roedel 2000

Mixing in the Atmosphere

Meridional (E-W), zonal (N-S), and interhemispheric mixing times of the atmosphere (D.J. Jacob 1999)

Interhemispheric Exchange Time from SF6 Time Series

Maiss and Levin 1994, GRL 21(7), 569-572

SF6 Global Atmospheric Increase

courtesy Levin & Engel

Summary

- The Atmospheric Dynamics is largely described by the Navier-Stokes Equations in a rotating coordinate system
- Frontal zones and fronts are an important phenomenon in the Earth‘s atmosphere
- Fronts are strongly tilted Formula of Margules
- High- and low pressure systems form in the descending and ascending branch of baroclinc waves

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