The simplifed momentum equations Height coordinatesPressure coordinates

fvz

uK

x

P

dt

du

2

21

fuz

vK

y

P

dt

dv

2

21

TR

Pg

z

P

d

fvx

P

dt

du

1

fuy

P

dt

dv

1

The simplifed momentum equations

fvxdt

du

fuydt

dv

fvz

uK

xdt

du

2

2

fuz

vK

ydt

dv

2

2

p

TR

pd

Height coordinates Pressure coordinates

In this section we will derive the four most important relationships describing atmospheric structure

1. Hydrostatic Balance: the vertical force balance in the atmosphere

2. The Hypsometric Equation: the relationship between the virtual temperature of a layer and the layer’s thickness

3. Geostrophic Balance: the most fundamental horizontal force balance in the atmosphere

4. Thermal wind Balance: a relationship between the wind at an upper level of the atmosphere and the temperature gradient below that level

We will then expand our understanding of balanced flow to include more complicated force balances in horizontal flows

1. Geostrophic Flow: the flow resulting from a balance between the PGF and Coriolis force in straight flow

2. Inertial Flow: curved flow resulting when the Coriolis force balances the “Centrifugal” force

3. Cyclostrophic Flow: Flow that results when the pressure gradient force balances the Centrifugal force

4. Gradient flow: Flow that results when the pressure gradient force and Coriolis force balance the Centrifugal force

Virtual Temperature: The temperature that a parcel of dry air would have if it were at the same pressure and had the same density as moist air.

Derivation:Start with ideal gas law for moist air:

TRTRP vvdd Now treat moist air as if it were dry by introducing the virtual temperature Tv

vdvdvdvvdd TRTRTRRP

What is the relationship between the temperature, TAnd the virtual temperature Tv?

P = pressured = dry air densityv = vapor density= air densityR= gas constantRv = vapor gas constantRd = dry air gas constantT = Temperature

RTP

Write:V

M

Multiply and divide second term by Rd

Cancel Rd and rearrange

vdvdvvdd TRTRR

T

VM

VM

RR

VM

VM

Tvd

d

vvd

v

M = mass of airMd = mass of dry airMv = mass of vaporV = volume

vdvd

vv

dd TR

V

M

V

MTR

V

MR

V

M

vdvd

d

vdvd

d TRV

M

V

MT

R

RR

V

MR

V

M

From last page:

Cancel V, and divide top and bottom terms by Md:

T

VM

VM

RR

VM

VM

Tvd

d

vvd

v

T

MM

RR

MM

T

d

v

d

v

d

v

v

1

1

Introduce mixing ratio: rv = Mv/Md and let = Rd/Rv

Tr

rT

v

v

v

1

11

Tr

rT

v

v

v

1

11

From last page:

Approximate (1+rv)-1 = 1- rv and use 1/ = 1.61

TrrrTrrT vvvvvv261.161.1161.111

Neglect term2vr

TrT vv 61.01

HYDROSTATIC BALANCE

gz

P

1

0

The atmosphere is in hydrostatic balance essentially everywhere except in core regions of significant storms such as hurricanes and thunderstorms

Consider a column of atmosphere that is 1 m by 1 m in area and extends from sea level to space

The Hypsometric Equation

Let’s isolate the part of this column that extends between the 1000 hPa surface and the 500 hPa surface

How much mass is in the column?

kgsm

mhPa

mNhPaMass 04.5102

81.9

11

100)5001000(

22

2

How thick is the column? That depends…………

gz

P

Hydrostatic equation

Ideal Gas LawvdTRp

Substitute ideal gas law into hydrostatic equation

p

p

g

TRz vd

Integrate this equation between two levels (p2, z2) and (p1, z1)

2

1

2

1

p

p

vd

z

z p

p

g

TRz

2

1

2

1

p

p

vd

z

z p

p

g

TRz

Problem: Tv varies with altitude. To perform the integral on the right we have to consider the pressure weighted column average virtual temperature given by:

1

2

2

1

lnp

p

vd

z

z

pg

TRz

From previous page

1

2

1

2

ln

ln

p

p

p

p

v

v

p

pT

T

We can then integrate to give

2

112 ln

p

p

g

TRzz vd

This equation is called the Hypsometric Equation

The equation relates the thickness of a layer of air between two pressure levels to the average virtual temperature of the layer

Geopotential Height

We can express the hypsometric (and hydrostatic) equation in terms of a quantity called the geopotential height

Geopotential (): Work (energy) required to raise a unit mass a distance dz above sea level

gdzd

2

1lnp

pTR vd

Meteorologists often refer to “geopotential height” because this quantity is directly associated with energy to vertically displace air

Geopotential Height (Z):

00 g

gz

gZ

g0 is the globally averaged

Value of gravity at sea level

For practical purposes, Z and z are about the same in the troposphere

Atmospheric pressure varies exponentially with altitude, but very slowly on a horizontal plane – as a result, a map of surface (station) pressure looks like a map of altitude above sea level.

variation of pressurewith altitude

observed surface pressureover central North America

Station pressures are measured at locations worldwide

Analysis of horizontal pressure fields, which are responsible forthe earth’s winds and are critical to analysis of weather systems,requires that station pressures be converted to a common level,

which, by convention, is mean sea level.

Reduction of station pressure to sea level pressure:

dzRT

g

p

dp

STA

SL

STA z v

p

p

0Integrate hypsometric equation

sea level pressure (PSL) to station pressure (PSTN)and right side from z = 0 to station altitude (ZSTN).

(4)

STNvd

STNSL zTR

gpp lnln (5)

BUT WHAT IS Tv? WE HAVE ASSUMED A FICTICIOUSATMOSPHERE THAT IS BELOW GROUND!!!

National Weather Service Procedure to estimate Tv

1. Assume Tv = T 2. Assume a mean surface temperature = average of

current temperature and temperature 12 hours earlierto eliminate diurnal effects.

3. Assume temperature increases between the station and sea level of6.5oC/km to determine TSL.

4. Determine average T and then PSL.

In practice, PSL is determined using a table of “R” values, whereR is the ratio of station pressure to sea level pressure, and the tablecontains station pressures and average temperatures.

Table contains a “plateau correction” to try to compensate for variations in annual mean sea level pressures calculated for nearby stations.

Implications of hypsometric equation

2

1lnp

p

g

TRz vd

Consider the 1000—500 hPa thickness field. Using11287 KkgJRd

281.9 smg

metersTnessLayerThick v3.20

metersTnessLayerThick v 3.20

Ballpark number: A decrease in average 1000-500 hPa layer temperature of 1K leads to a reduction in thickness of the layer of 60 hPa

Implications of Hypsometric Equation

A cold core weather system (one which has lower temperature at its center) will winds that increase with altitude

How steep is the slope ofThe 850 mb pressure surface in the middle latitudes?

1630 meters @ 40oN

1200 meters @ 80oN

40o 60 nautical miles/deg 1.85 km/nautical mile = 4452 km

Slope = 430 m/4452000 m ~ 1/10,000 ~ 1 meter/10 km

How steep is the slope ofThe 250 mb pressure surface in the middle latitudes?

10,720 meters @ 40oN

9,720 meters @ 80oN

40o 60 nautical miles/deg 1.85 km/nautical mile = 4452 km

Slope = 1000 m/4452000 m ~ 1/4,000 ~ 1 meter/4 km

GEOSTROPHIC BALANCE

fvx

p

dt

du

1

0

fuy

p

dt

dv

1

0

A state of balance between the pressure gradient force and the Coriolis force

Air is in geostrophic balance if and only if air is not accelerating (speedingup, slowing down, or changing direction).

For geostrophic balance to exist, isobars (or height lines on a constant pressurechart) must be straight, and their spacing cannot vary.

Geostrophic balance

x

p

fvg

1

y

p

fug

1

Geostrophic wind

The wind that would exist if air was in geostrophic balance

The Geostrophic wind is a function of the pressure gradient and latitude

yfug

1

xfvg

1

In pressure coordinates, the geostrophic relationships are given by

Where on this map of the 300 mb surface is the air in geostrophic balance?

Geostrophic Balance and the Jetstream

yfug

1

xfvg

1

Take p derivative:

Pyfp

ug 1

Pxfp

vg 1

Substitute hydrostatic eqn

y

T

fp

R

p

udg

p

TR

Pd

x

T

fP

R

P

vdg

THE RATE OF CHANGE OF THE GEOSTROPHIC WIND WITHHEIGHT (PRESSURE) WITHIN A LAYER IS PROPORTIONAL TO THE HORIZONTAL TEMPERATURE GRADIENT WITHIN THE LAYER

y

T

fp

R

p

udg

x

T

fP

R

P

vdg

We can write these two equations in vector form as

Tkfp

R

p

Vdg

or alternatively

Pf

k

p

Vg

The “Thermal Wind” vector

The “Thermal Wind” is not a wind! It is a vector that is parallel to the mean isotherms in a layer between two pressure surfaces and its magnitude is proportional to the thermal gradient within the layer.

When the thermal wind vector is added to the geostrophic wind vector at a lower pressure level, the result is the geostrophic wind at the higher pressure level

A horizontal temperaturegradient leads to a greaterslope of the pressure surfaces above the temperature gradient.

More steeply sloped pressuresurfaces imply that a strongerpressure gradient will existaloft, and therefore a strongergeostrophic wind.

Note the position of the front at 850 mb……

….and the jetstream at 300 mb.

Implications of geostrophic wind veering (turning clockwise) and backing (turning counterclockwise) with height

Winds veering with height – warm advection

Cold air

Warm air

Implications of thermal wind relationship for the jetstream

Note the position of the jet over the strongest gradient in potential temperature (dashed lines) (which corresponds to the strongest temperature gradient)

Implication of the thermal wind equation for the general circulation

Upper level winds must be generally westerly in both hemispheres since it is coldest at the poles and warmest at the equator

Natural Coordinates and Balanced Flows

To understand some simple properties of flows, let’s consider atmospheric flow that is frictionless and

horizontalWe will examine this flow in a new coordinate system called “Natural Coordinates”

In this coordinate system:the unit vector is everywhere parallel to the flow and positive along the flowthe unit vector is everywhere normal to the flow and positive to the left of the flow

i

n

iVV

The velocity vector is given by:

The magnitude of the velocity vector is given by: where s is the measure of distance in the direction. dt

dsV

i

The acceleration vector is given by:dt

idV

dt

Vdi

dt

Vd

where the change of with time is related to the flow curvature.i

dt

idV

dt

Vdi

dt

Vd

We need to determine dt

id

The angle is given by ii

i

R

s

Where R is the radius of curvature following parcel motion

0R

0R

n

directed toward center of curve (counterclockwise flow)

n

directed toward outside of curve (clockwise flow)

iR

s

Rs

i 1

nRds

id

s

i

s

1

0

lim

Note here that points in the positive direction in the limit that approaches 0.

i n

s

nR

VV

R

n

dt

ds

ds

id

dt

id

R

Vn

dt

Vdi

dt

Vd 2

dt

idV

dt

Vdi

dt

Vd

Let’s now consider the other components of the momentum equation, the pressure gradient force and the Coriolis Force

Coriolis Force: nfVVkf

Since the Coriolis force is alwaysdirected to the right of the motion

nn

isp

PGF: Since the pressure gradient forcehas components in both directions

The equation of motion can therefore be written as:

nfVnn

is

nR

Vi

dt

Vd

2

nfVnn

is

nR

Vi

dt

Vd

2

Let’s break this up into component equations:

sdt

dV

02

n

fVR

V

To understand the nature of basic flows in the atmosphere we will assume that the speed of the flow is constant and parallel to the height contours so that

0

sdt

dV

Under these conditions, the flow is uniquely described by the equation in the yellow box

02

n

fVR

V

Centrifugal Force

Coriolis Force

PGF

Geostrophic Balance: PGF = COR

Inertial Balance: CEN = COR

Cyclostrophic Balance CEN = PGF

Gradient Balance CEN = PGF + COR

02

n

fVR

V

Geostrophic flow

Geostrophic flow occurs when the PGF = COR, implying that R

For geostrophic flow to occur the flow must be straight and parallel to the isobars

nfVg

1

Pure geostrophic flow is uncommon in the atmosphere

02

n

fVR

V

Inertial flow

Inertial flow occurs in the absence of a PGF, rare in the atmosphere but common in the oceans where wind stress drives currents

f

VR

This type of flow follows circular, anticyclonic paths since R is negative

Time to complete a circle:

sinsin2

2222

ffR

R

V

Rt

is one half rotation is one full rotation/day

sin

5.0

sin

day

called a half-pendulum day

daysday

23.213sin

5.0

Power Spectrum of kinetic energy at 30 m in the ocean near Barbados (13N)

02

n

fVR

V

Cyclostrophic flow

Flows where Coriolis force exhibits little influence on motions (e.g. Tornado)

nR

V

2

nR

V

2

2

1

n

RV

2

1

n

RV

Cyclostrophic flows can occur when the Centrifugal force far exceeds the Coriolis force

fR

V

fV

RV

/2

, a number called the Rossby number

A synoptic scale wave: 1.01010

10614

1

ms

sm

fR

V

A tornado: 10001010

100314

1

ms

sm

fR

V

NO

YES

Venus (rotates every 243 days): 1001010

100717

1

ms

sm

fR

VYES

2

1

n

RV

In cyclostrophic flow around a low, circulation can rotate clockwise or counterclockwise (cyclonic and anticyclonic tornadoes and smaller vortices are observed)

WHAT ABOUT A HIGH???

Venus has a form of cyclostrophic flow, but more is going on that we don’t understand

02

n

fVR

V Gradient flow

2/1222

422

14

n

RRffR

R

nRff

V

This expression has a number of mathematically possible solutions, not all of which conform to reality

R is the radius of curvature following parcel motion

0R

0R

n

directed toward center of curve (counterclockwise flow)

n

directed toward outside of curve (clockwise flow)

is the height gradient in the direction ofn

is the geopotential height

n

n

the unit vector is everywhere normal to the flow and positive to the left of the flow

V is always positive in the natural coordinate system

2/122

42

n

RRffR

V

0,0

Rn

Solutions for For radical to be positive

nR

Rf

4

22

Therefore: must be negative.

2/122

42

n

RRffR

V = negative = UNPHYSICAL

2/122

42

n

RRffR

V

0,0

Rn

Solutions for Radical >

2

fRPositive root physical

Anticyclonic 0R

n

outward

Increasing in direction (low) n

Negative root unphysical

Called an “anomalous low” it is rarelyobserved (technically since f is never 0 inmid-latitudes, anticyclonic tornadoes areactually anomalous lows

2/122

42

n

RRffR

V

0,0

Rn

Solutions for Radical >

2

fRPositive root physical

Cyclonic 0R

n

inward

decreasing in direction (low) n

Negative root unphysical

Called an “regular low” it is commonlyobserved (synoptic scale lows to cyclonicallyrotating dust devils all fit this category

2/122

42

n

RRffR

V

0,0

Rn

Solutions for Radical >

2

fRPositive root physical

Cyclonic 0R

n

inward

decreasing in direction (low) n

Negative root unphysical

Called an “regular low” it is commonlyobserved (synoptic scale lows to cyclonicallyrotating dust devils all fit this category

2/122

42

n

RRffR

V

0,0

Rn

Solutions for

Antiyclonic 0R

n

outward

decreasing in direction (high) n

nR

Rf

4

22

or radical is imaginary

therefore2

fRV or fV

R

V

2

2

Called an “anomalous high” : Coriolis force never observed to be < twice the centrifugal force

Positive Root

2/122

42

n

RRffR

V

0,0

Rn

Solutions for

Antiyclonic 0R

n

outward

decreasing in direction (high) n

nR

Rf

4

22

or radical is imaginary

Called a “regular high” : Coriolis force exceeds the centrifugal force

Negative Root

Condition for both regular and anomalous highs

04

22

n

RRf

nR

Rf

4

22

4

2 Rf

n

This is a strong constraint on the magnitude of the pressure gradient force in the vicinity of high pressure systems

Close to the high, the pressure gradient must be weak, and must disappear at the high center

Note pressure gradients in vicinity of highs and lows

The ageostrophic wind in natural coordinates

02

n

fVR

V

nfVg

1

02

gVVfR

V

fR

V

V

Vg 1

For cyclonic flow (R > 0) gradient wind is less than geostrophic wind

For anticyclonic flow (R < 0) gradient wind is greater than geostrophic wind

gVV

gVV gVV

Convergence occurs downstream of ridges and divergence downstream of troughs