Potential temperature
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Initial state: Ti, P |
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Ti |
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P 0,286 |
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Terminal state: |
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Ti0 |
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θ. 1000 hPa |
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P0 |
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0,286 |
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Ti |
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P |
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1000 |
0,286 |
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Ti |
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P |
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1000 |
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0,286 Rc |
cp |
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Properties of the potential temperature
1. It remains unchanged at adiabatic displacement (up or down)
ln lnT Rc ln1000 Rc ln P |
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cp |
cp |
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ln lnT Rc ln1000 |
Rc ln P |
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dP |
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cp |
cp |
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RcTi |
P cpdTi |
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d |
dTi |
Rc dP |
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Constant |
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cpTi |
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p |
P |
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d |
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dTi Rc dP 0 |
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d 0 |
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Ti |
cp P |
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If an air particle moves without heat exchange with environment, its potential temperature remains unchanged, while its molecular temperature changes.
Any variation of the potential temperature means that the air gains or looses energy.
dq cpTi d
12
2. Potential temperature determines total amount of energy.
d |
dTi |
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Rc dP |
cpTi |
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dP |
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Ti |
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cp P |
Ticp |
cpdTi RcTi P |
dq |
g Ti Te
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dq c T |
d |
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RcTi |
dP |
RcTi g |
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Pdz |
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gTi |
dz |
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P |
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Te Rc P |
Te |
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p i |
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T c |
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d |
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dT gTi |
dz |
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p |
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p |
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T |
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e |
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dz gdz gdz |
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gdz g Ti |
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1 dz gdz g Ti Te |
dz |
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T |
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T |
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e |
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e |
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с T |
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Te |
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dT gdz g Ti |
dz |
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p i |
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p i |
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T |
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d |
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Ti |
Te |
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сpTi |
cpdTi gdz g |
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dz |
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dЭi |
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dФ |
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Ei is instability energy |
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Эi cpTi const |
dEi |
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(CAPE=convective |
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available potential |
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Entalpy or total heat |
Ф is potential |
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energy) |
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energy |
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(Geopotential) |
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d |
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сpTi |
dЭi dФ dEi |
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сpTi |
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dПi |
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Total energy of a particle of a unit mass |
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dПi dЭi dФ dEi |
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d |
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dПi |
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Potential temperature variation is uniquely |
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cpTi |
determined by the total energy variation of the air |
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particle. |
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At adiabatic displacement, the total energy of the air particle remains unchanged. 14
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Z |
2 |
dE |
Z 2 |
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E |
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g Ti |
Te dZ |
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E |
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g Ti |
Te Z |
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E g Ti Te Z dW |
Z |
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T |
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dt |
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For a unit of mass m=1 kg
F a m a
E F L aL
In our case L=∆Z
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a dW |
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W W Ei |
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dt |
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Ti |
Te |
dW |
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g |
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dt |
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Wmax 2Ei |
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15
Criterion of the atmosphere stability
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Te |
Environmental vertical temperature gradient or |
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Z |
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Ti Ti0 a z z0 Ti0 0,01 z z0 |
lapse rate |
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Temperature and other meteorological parameters distribution with height is known as stratification of the atmosphere.
Z |
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Z2 |
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Z |
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e2 |
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e1 |
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Z2 |
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Case 1 |
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a |
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Case 2 |
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Zo |
e1 a |
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Zo |
e1 a |
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Z1 |
T |
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Z1 |
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T |
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Ti Te |
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Ti |
Te |
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Case 3 |
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Ti Te |
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Ti Te |
Z |
Ti Te |
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Ti |
Te |
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a e3 |
Z2 |
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The atmosphere is |
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The atmosphere is |
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stable (stable |
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unstable (unstable |
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Zo |
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stratification) |
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stratification) |
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Z1 |
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dW |
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T |
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Neutral state |
Ti Te |
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dt |
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Te |
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The reasoning above suggests: |
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The atmosphere is unstable |
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The atmosphere is stable |
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The atmosphere is in the neutral state |
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Our reasoning and conclusion are valid assuming the process goes without any exchange with environment. In the real atmosphere at leastways some exchange takes place. However, in spite this exchange, conclusion on stability conditions and corresponding vertical displacement are valid.
Potential temperature variation at different types of the stratification
Case 1. Unstable a
stratification
P |
P2 |
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P1 |
P=1000 hPa |
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2 |
T |
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Potential temperature decreases with height in unstable atmosphere or if potential temperature decreases with height, it means that the atmosphere is unstable
Case 2. Stable |
P |
P2 |
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0 |
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stratification |
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P1 |
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1 |
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T |
P=1000 hPa |
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Potential temperature increases with height in the stable atmosphere or if potential temperature increases with height, it means that the atmosphere is stable
From the similar reasoning it goes without saying that at the neutral condition potential temperature does not vary with height.
0Z
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Theoretical support of the reasoning above
1000 |
R c p |
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ln lnT |
R |
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ln1000 |
R |
ln P |
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P |
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cp |
cp |
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1 1 T 
R P
z T z 
cp P z
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1 g |
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z |
T |
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T cp |
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Unstable state |
a 0 |
Stable state |
a 0 |
Neutral state |
a 0 |
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R P |
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g g |
R P |
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Pcp z |
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cp P RT |
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Pcp |
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T 0 |
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z |
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All above refers to the dry air (non-saturated air)
Very hot surface
In case of very dry air, instead of cloud formation, overturning of layer occurs. Cold air moves down, and the overheated air ascents up in convective flows.
This process results in formation of eddies similar to tornado, but these vortexes are very small. In our latitudes, this process leads to gusty winds and squalls.
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