Atmospheric Dynamics Leila M. V. Carvalho Dept. Geography, UCSB.

Atmospheric Dynamics

Leila M. V. CarvalhoDept. Geography, UCSB

Review: Kinematic of the horizontal flowStreamlines: lines parallel to the horizontal velocity V at a particular level and at a

particular instant in timehttp://weather.unisys.com/surface/sfc_con_stream.html

Natural Coordinates:

Y

X

n n

ss

n and s are natural coordinates (perpendicular and parallel to the flow

Definitions

Sheared with no curvature, no diffluence, stretching or divergence

Rotation with cyclonic curvature (NH) and cyclonic shear, no diffluence or stretching (and divergence

Radial flow with velocity directly proportional to radius. Diffluence, stretching, divergence and NO CURVATURE (or vorticity)

Hyperbolic flow: difluence and straching, no divergence (terms cancel). Shear and curvature cancel (vorticity free)

What is going on here?

Y

X

n n

ss

Forces in the Atmosphere

• Equation of motion: (First and second Laws of Newton)

• Real forces (independent on the rotating system): gravity, pressure gradient force and frictional force

• Apparent forces due to rotation: apparent centrifugal force (affects gravity) and Coriolis (correction for horizontal movements).

Apparent forces:

• Centrifugal force:

Where RA is vector perpendicular to axis of rotation and is angular velocity of earth

Combine with gravity to define "effective" gravity

Coriolis force:

• Coriolis force takes care of rotational effects caused by motion relative to surface

Ω At rest over the Earth surface will have cetrifugal acceleration= Ω2R.

Suppose it moves eastward with speed u: the centrifugal force would increase to:Centrifugal force=

RR

u2

R

Coriolis force:

2

22

22

R

Ru

R

uRRR

R

u

Expanding the equation we have now:

Centrifugal force due to rotation of the Earth (independent of the relative velocity

Deflecting forces that act outward along the vector R

Synoptic scale motions u<< ΩR:Last term can be neglected in a first approximation

Coriolis Force

Coriolis force can be divided into vertical and meridional components :

R

φR

uR2φ

cos2 u

sin2 uA relative motion along the east-west coordinate will produce an acceleration in the north-south direction given by:

sin2 udt

dvCo

And vertical acceleration given by: cos2 udt

dwCo

To the right of the movement in the NH

Suppose now that a particle initially at rest on the Earth is set in motion equatoward by

impulsive forcesΩ

RAs it moves equatorward it will conserve its angular momentum in the absence of torques: a relative westward velocity must develop

R + δRIf we expand the right hand side and neglect second order differentials (and assume that δR<<R and solve for δu, we get:

oaRu sin22

a

a= Earth’s Radius

oo vdt

da

dt

du sin2sin2

dt

dav

Northward velocity component

Real forces in the Atmosphere

Pressure gradient Force

Low Pressurep2

High Pressurep1

wind direction

Pressure Gradient

REMEMBER THAT A “GRADIENT” ALWAYS POINT TOWARD THE HIGHEST MAGNITUDES OF THE SCALAR.

Pressure Gradient Force

x

pP 1

)8.7(1

;1

y

pP

x

pP yx

Hydrostatic Equation:

gz

p

gdzd

Definition of Geopotential Geopotential Height

z

oo

gdzgg

zZ

0

1)(

ZgzgpP o1

z

y

>0 for sure

See Holton, 1979, second Ed. Chap1, pg. 21

ZgzgpP o1

Surface of constant Pressure

Changes in geopotential height

1

2

12

p

p

vo

d

p

dpT

g

RZZ

Changes in geopotential height imply in the existence of pressure gradient forces

Zgo

ppzppz yy

zg

y

p

xx

zg

x

p

1

;1

Winds and geopotential height: example: sea breeze

ZgPForceGradient o:

Z

LAND OCEAN

High Pressure

W E

ppz xx

zg

x

p

1

Friction or Viscosity Force

zF

1 τ is the shear stress and is the

rate of vertical exchange of horizontal momentum N/m2

τs at the surface

Friction or Viscosity Force

zF

1 τzx is the shear stress in the

horizontal direction x due to the stress acting vertically

z

uzx

subscripts indicate that τzx is the shear stress in x direction due to vertical shear and μ is the dynamic viscosity coefficient

In summaryFriction

Very small outside boundary layerDepends on vertical gradient in vertical component of shear stress

•Actual processes very complex, with turbulence playing key role•Approximate shear stress in surface boundary layer:•Shear stress depends on strength of vertical shear in horizontal wind. Empirically:

ν = viscosity coefficient = μ/ρ ~10-5m2s-1

Drag coefficient, CD, depends on surface roughness and static stability

Horizontal Equation of MotionNewton’s Law in vectorial form per unity of mass:

FVkV

fpFCPdt

d

1

In a tangent plan we have (this is important to remember):

xFfvx

p

dt

du

1

yFfuy

p

dt

dv

1

Remember that Friction is defined as a negative component that is supposed to decrease (decelerate) the speed

We can eliminate density by using the relationship between pressure gradient and geopotential

FVkV

fdt

d

Test your understanding:

Estimate pressure gradient and, coriolis parameter in Kansas ~38oNRepresent winds around the Low and High pressure systems

Geostrophic windBy using scale analysis of horizontal equations it can be shown that:

Horizontal velocity scale:

Length scale:

Depth scale:

Horizontal pressure fluctuation scale:

Time scale (advective):

Coriolis Scale:

In the free atmosphere Coriolis balances with Gradient Force

FVkV

fdt

d

Friction effect:

Geostrophic winds

5460m

5560m

5640m

L

H

FGP

CF

CF CF

FGP

CF

wind

FGP

FGPFGP

Estimate the geostrophic winds given this distribution of geopotential height, assuming that the spatial interval between the two lines is equal 100km. Assume this region is in midlatitudes of the NH

kVfg

1

yfug

1

xfug

1

Tridimensional view

Northern Hemisphere

Gradient WindCurved trajectories when the wind direction is changed: the centripetal (or centrifugal) acceleration needs to be considered in the balance of forcesCentripetal acceleration is given by: V2/RT, where RT is the local radius of curvature of the air trajectories.

FVkV

fdt

dVkn f

R

V

T

2

The signs of these terms depend on the curvature

The centrifugal force Acts in the same direction as Coriolis

Since Co is dependent on the wind speed, and since the centrifugal force is in the same direction as Co, the balance of forces can be achieved at slower speeds compared with a geostrophic one : SUBGEOSTROPHIC

The centrifugal force is opposite to Co: Balance is achieved at higher speeds compared with the geostrophic balance:SUPERGEOSTROPHIC WINDS

Where would you expect to observe geostrophic balance, gradient balance, supergeostropic and subgeostrophic winds? Show the balance of forces in these

regions http://www.opc.ncep.noaa.gov/Loops/pac500satf00/Pacific_500mb_Analysis_07_Day.shtml

Thermal Wind

• Is not an actual wind, as it does not blow the dust from the ground or rocks the leaves in the trees.

• The purpose of thermal winds is to indicate a relationship between vertical shear in the geostrophic wind and temperature gradient that will help in weather forecast.

• It can be obtained by writing the geostrophic equation for two different pressure surfaces and subtracting them (to calculate the shear in the intervening layer)

1212

1 kVV

fgg

x

z2

1 1212ZZ

f

gogg kVV

In terms of geopotential height

• In component form:

y

ZZ

f

guu o

gg

1212

x

z2

1 x

ZZ

f

gvv o

gg

1212

• In other words, the thermal wind equation states that the vertical shear of the geostrophic wind within the layer between any two pressure surfaces is related to the horizontal gradient of thickness

Interpretation

xx

z

1212 gggg VVVV

12 gg VV

The wind shear (red arrow)

012 gg uu

012 gg vv

In this example in the NH, f >0 the atmospheric thickness is decreasing or increasing as we move north? Is it increasing or decreasing as we move westward?

• In component form:

y

ZZ

f

guu o

gg

1212

x

ZZ

f

gvv o

gg

1212

The wind shear (red arrow)

012 gg uu

012 gg vv

Answer: thickness decreases northward and eastward

The thermal wind is parallel to thickness contours

Relationships with horizontal temperature gradient

Tp

p

f

Rgg

kVV

2

112

ln

• Barotropic atmosphere: density depends only on the pressure (isobaric surfaces are also surfaces of constant density). Isobaric surfaces will be also isothermals (law of gases ):

0ln

0

p

VT g

p

• Geostrophic winds is independent of height in a barotropic atmosphere (geopotential heights are stacked on the top of one another like dishes)

Baroclinic atmosphere:

• Baroclinic atmosphere: Density depends on both the temperature and pressure. In a baroclinic atmosphere the geostrophic wind generally has vertical shear related to the horizontal temperature gradient by the thermal wind equation.

Equivalente Barotropic:

• Horizontal temperature gradients are such that the thickness contour are parallel to the geopotential height contours. In this case, the thermal wind equation states that the wind shear should be parallel to the wind itself: there is no change in direction of the wind.

Exercise 7.3• During the winter in the troposphere ~ 30oN, the zonally averaged

temperature gradient is ~0.75K per degree of latitude and the zonally averaged component of the geostrophic wind at the Earth’s surface is close to zero. Estimate the mean zonal wind at the jet stream level ~ 250hPa

Solution: take the zonal component and average:

y

TRuu gg

250

1000ln

sin21000250 T

p

p

f

Rgg

kVV

2

112

ln

1515

11

2508.36

1011.1

75.04ln

30sin1029.72

deg287

ms

m

Ku

og s

kgJ

Cold and warm temperature advection

• Cold advection: flow across the isotherms from a colder to a warmer regions

• Warm advection (opposite)• The thermal wind theory tells us that VT ‘blows’ parallel to the

thickness with the colder (warmer) air to the left of the wind in the NH (SH)

• If you know the geostrophic wind (the one that can blow your hair) between two levels you can estimate the mean wind direction in that layer.

• Joining both info will tell you if the present wind configuration will advect cold or warm air, and therefore, you can use that to forecast the weather!

Discussion: are the regions marked in the map cooling or warming and why?

2 3

1 5

2 31 5

Thickness (1000-500) 700hPa height/temperature/winds)

2 31

4

Discuss the advection of temperature in the regions marked with a star . Plot the thermal wind, the temperature gradient (vectors). Assume that the 700hPa winds represent the mean wind between 1000 and 500 hPa.

925 mb

700 mb

850mb

500mb