### thermal wind balance

```g Z
f y
Z
RT
( 2)

p
gp
(1) u g  
1.4 thermal wind balance
geostrophic wind
hypsometric eqn
ug
plug (2) into (1)
 RT 

u g g 
gp 

p
f
y
R T

fp y
finite difference expression:
R p T
u g 
f p y
ug
greater
thickness
ug=0
lower
thickness
y
this is the thermal wind: an increase in wind with height due to a
The thermal wind blows ccw around cold pools in the same way as the geostrophic wind blows
ccw around lows. The thermal wind is proportional to the T gradient, while the geostrophic
wind is proportional to the pressure (or height) gradient.
Let’s verify qualitatively that climatological temperature and wind fields are
roughly in thermal wind balance.
For instance, look at the meridional variation of temperature with height (in Jan)
Around 30-45 ºN, temperature drops northward, therefore westerly
winds increase in strength with height.
thermal wind
The meridional temperature gradient is
large between 30-50ºN and 1000-300 hPa
Therefore the zonal wind increases
rapidly from 1000 hPa up to 300 hPa.
Question:
Why, if it is colder at higher latitude, doesn’t the wind
continue to get stronger with altitude ?
There is definitively a jet ...
Answer: above 300 hPa, it is no longer colder at
higher latitudes...
 pT
tropopause
 pT
 pT
 pT
Z500
v g 500
g ˆ
 k  Z 500
f
Z500-Z1000
vg
 u g v g  R  T T  R
R ˆ
   
 
 
,
,

k  T
 p   p  p  fp  y x  fp fp
g
g
vT 500,1000  v g 500  v g1000  kˆ  Z 500  Z1000   kˆ  Z
f
f
baroclinicity
•
•
The atmosphere is baroclinic if a horizontal temperature gradient is present
The atmosphere is barotropic if NO horizontal temperature gradient exists
– the mid-latitude belt typically is baroclinic, the tropical belt barotropic
•
The atmosphere is equivalent barotropic if the temperature gradient is
aligned with the pressure (height Z) gradient
– in this case, the wind increases in strength with height, but it does not change
direction
equivalent barotropic
baroclinic
geostrophic
wind at
various
levels
cold
Z
warm
T
height temperature
warm
 

vT  vg 2  vg1

vg1

vg1
cold
1.4.2 Geostrophic T advection:
cold air advection (CAA) & warm air advection (WAA)
highlight areas of cold air advection (CAA) & warm air
WAA
CAA
WAA & CAA
the solenoid method
geostrophic wind:

g
g  Z Z 
vg  kˆ  Z  
,
f
f  y x 

g
vg  Z
f
geo. temperature advection is:

 vg  T
the magnitude is:

g
vg  T  Z T
f
the smaller the box, the
stronger the temp advection
fatter arrow: larger T gradient
T
Thermal wind and geostrophic temperature advection
Let us use the natural coordinate and choose the s direction along
the thermal wind (along the isotherms) and n towards the cold air.
Rotating the x-axis to the s direction, the advection equation is:
local T
change
∂T
∂T
 Vn
,
∂
t
∂
n
(not et hat
∂T
 0)
∂
s
where Vn is the average wind speed perpendicular to the thermal wind.
The sign of Vn
cold
warm
 

vT  vg 2  vg1
VT
Vn
warm
+
VT
Vn
cold
-
Thermal wind and temperature advection
∂T
∂T
 Vn
,
∂
t
∂
n
∂T
(not et hat
 0)
∂
s
VT
VT
Vn
∂T
>0
∂
t
WAA
+
Vn
-
∂T
<0
∂
t
CAA
If the wind veers with height, Vn is positive and there is warm advection.
If the wind is back with height, Vn is negative and there is cold advection.
thermal wind and temperature advection
Procedure to estimate the temperature advection in a layer:
1. On the hodograph showing the upper- and low-level wind, draw the
thermal wind vector.
2. Apply the rule that the thermal wind blows ccw around cold pools, to
determine the temperature gradient, and the unit vector n (points to
cold air)
3. Plot the mean wind Vn , perpendicular to the thermal wind. Note
that Vn is positive if it points in the same direction as n. Then the
wind veers with height, and you have warm air advection.
If there is warm advection in the lower layer, or cold advection in the
upper layer, or both, the environment will become less stable.
example
y
 

vT  vg 850  vg1000

vg1000
Vn
Vn  0
n
s

vg 850
x
∂T
>0
∂
t
veering wind  warm air advection
between 1000-850 hPa
friction-induced
near-surface
convergence into lows/trofs
1.5 vorticity
shear and curvature vorticity
a    f

 a or  ?
a

 v   a
Hovmoller diagrams (Fig. 1.20)
time scales of atmospheric variability
Lovejoy 2013, EOS
time scales of atmospheric variability
Lovejoy 2013, EOS
(1) Scales of atmospheric motion
Note two spectral extremes:
[shifted x10 to right]
(a) A maximum at about 2000 km
(b) A minimum at about 500 km
SD  k
5
3
inertial
subrange
1000
Gage and Nastrom (1985)
100
10
wavelength [km]
1
synoptic scale
Big whirls have little whirls
that feed on their velocity;
and little whirls have lesser whirls,
and so on to viscosity.
-Lewis Fry Richardson
FA=free atmos.
BL=bound. layer
L = long waves
WC = wave cyclones
TC=tropical cyclones
cb=cumulonimbus
cu=cumulus
CAT=clear air turbulence
From Ludlam (1973)
Scales of atmospheric motion
Markowski & Richardson 2010, Fig. 1.1
Scales of atmospheric motion
•
Air motions at all scales from planetary-scale to microscale explain weather:
– planetary scale: low-frequency (10 days – intraseasonal) e.g. blocking highs (~10,000
km) – explains low-frequency anomalies
size such that planetary vort adv > relative vort adv
• hydrostatic balance applies
•
–

 v g b   v g  
synoptic scale: cyclonic storms and planetary-wave features: baroclinic instability
(~3000 km) – deep stratiform clouds
smaller features, whose relative vort adv > planetary vort adv
• size controlled by b=df/dy
• hydrostatic balance applies
•
–
mesoscale: waves, fronts, thermal circulations, terrain interactions, mesoscale
instabilities, upright convection & its mesoscale organization: various instabilities –
synergies (100-500 km) – stratiform & convective clouds
time scale between 2p/N and 2p/f
• hydrostatic balance usually applies
•
–
2p/N ~ 2p/10-2 ~ 10 minutes
2p/f = 12 hours/sin(latitude) = 12 hrs at 90°, 24 hrs at 30°
microscale: cumuli, thermals, K-H billows, turbulence: static instability (1-5 km) –
convective clouds
Size controlled by entrainment and perturbation pressures
• no hydrostatic balance
•
```