fv z u K x P dt du 2 2 1 fu z v K y P dt dv 2 2 1 T R Pg z P d fv x P dt du 1 fu y P dt dv 1 The simplifed momentum equations fv x dt du fu y dt dv fv z u K x dt du 2 2 fu z v K y dt dv 2 2 p T R p d Height coordinates Pressure coordinates
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The simplifed momentum equations Height coordinatesPressure coordinates.
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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