EART164: PLANETARY
ATMOSPHERES
Francis Nimmo
F.Nimmo EART164 Spring 11
Next 2 Weeks – Dynamics
• Mostly focused on large-scale, long-term patterns of
motion in the atmosphere
• What drives them? What do they tell us about
conditions within the atmosphere?
• Three main topics:
– Steady flows (winds)
– Boundary layers and turbulence
– Waves
• See Taylor chapter 8
• Wallace & Hobbs, 2006, chapter 7 also useful
• Many of my derivations are going to be simplified!
F.Nimmo EART164 Spring 11
Key Concepts
•
•
•
•
•
Hadley cell, zonal & meridional circulation
Coriolis effect, Rossby number, deformation radius
Thermal tides
Geostrophic and cyclostrophic balance, gradient winds
Thermal winds
u
Ro 
2 L sin 
du
1 P

 2 sin v  Fx
dt
 x
u
g T

z
fT y
F.Nimmo EART164 Spring 11
2. Turbulence
F.Nimmo EART164 Spring 11
Turbulence
• What is it?
• Energy, velocity and lengthscale
• Boundary layers
Whether a flow is turbulent or not
depends largely on the viscosity
Kinematic viscosity n (m2s-1)
Dynamic viscosity h (Pa s)
nh/
Gas dynamic viscosity ~10-5 Pa s
Independent of density, but it does
depend a bit on T
F.Nimmo EART164 Spring 11
Reynolds number
• To determine whether a flow is turbulent, we
calculate the dimensionless Reynolds number
uL
Re 
n
• Here u is a characteristic velocity, L is a
characteristic length scale
• For Re in excess of about 103, flow is turbulent
• E.g. Earth atmosphere u~1 m/s, L~1 km
(boundary layer), n~10-5 m2/s so Re~108 i.e.
strongly turbulent
F.Nimmo EART164 Spring 11
Energy cascade (Kolmogorov)
Energy in (e, W kg-1)
ul, El
l
Energy viscously
dissipated (e, W kg-1)
• Approximate analysis (~)
• In steady state, e is constant
• Turbulent kinetic energy
(per kg): El ~ ul2
• Turnover time: tl ~l /ul
• Dissipation rate e ~El/tl
• So ul ~(e l)1/3 (very useful!)
• At what length does viscous
dissipation start to matter?
F.Nimmo EART164 Spring 11
Kinetic energy and lengthscale
• We can rewrite the expression on the previous
page to derive El ~ e 2 / 3l 2 / 3
• This prediction agrees with experiments:
F.Nimmo EART164 Spring 11
Turbulent boundary layer
• We can think of flow near a boundary as consisting of
a steady part and a turbulent part superimposed
• Turbulence causes velocity fluctuations u’~ w’
u (z )
z
+
•Vertical gradient in steady horizontal
velocity is due to vertical momentum
u’, w’ transfer
•This momentum transfer is due to
some combination of viscous shear and
turbulence
•In steady state, the vertical momentum
flux is constant (on average)
•Away from the boundary, the vertical
momentum flux is controlled by w’.
•So w’ is ~ constant.
F.Nimmo EART164 Spring 11
Boundary Layer (cont’d)
• A common assumption for
turbulence (Prandtl) is that
z
du
w' ~ z
dz
Note log-linear plot!
turbulent
viscous
• But we just argued that w’ was
constant (indep. of z)
• So we end up with u ~ ln z
• This is observed experimentally
• Note that there are really two
boundary layers
F.Nimmo EART164 Spring 11
3. Waves
F.Nimmo EART164 Spring 11
Atmospheric Oscillations
d 2z
 2   g    
dt
Altitude
Actual Lapse Rate
   T T


T
Colder
Adiabatic Lapse Rate
 ,T
z0
,T
Air parcel
d 2 z g   dT   dT  
  

  z
2
dt
T   dz   dz  a 
Warmer
d 2z
2



NB z
dt 2
Temperature
• E.g. Earth (dT/dz)a=-10 K/km,
dT/dz=-6K/km (say), T=300 K,
NB=0.01s-1 so period ~10 mins
 NB
2
g   dT  g 
  

T   dz  C p 
NB is the Brunt-Vaisala frequency
F.Nimmo EART164 Spring 11
Gravity Waves
z
l
Cooling & condensation
u
Neutral buoyancy

• Common where there’s topography
• Assume that the wavelength is set by the topography
• So the velocity
 NB
u
l
2
• You also get gravity waves
propagating upwards:
F.Nimmo EART164 Spring 11
Gravity Waves
Venus
Mars
• What is happening here?
F.Nimmo EART164 Spring 11
Overcoming topography
• What flow speed is needed to propagate over a
mountain?
d
u
KE 

z

PE   g dz 
gz dz
z
1

PE  d 2 g
2
z
1
1
d   dT
(from before)

T
1 2
u
2
• So we end up with: u   d
• The Sierras are 5 km high, NB~0.01s-1, so wind
speeds need to exceed 50 ms-1 (110 mph!)
2
2
NB
2
F.Nimmo EART164 Spring 11
Rossby (Planetary) Waves
• A result of the Coriolis
acceleration 2 x u
• Easiest to see how they work
near the equator:
l
y
u
equator
•
•
•
•
Magnitude of acceleration ~ -2 u y/R (why?)
So acceleration a – displacement (so what?)
1/ 2
This implies wavelength l ~ uR / 
What happens if the velocity is westwards?
F.Nimmo EART164 Spring 11
Kelvin Waves
• Gravity waves in zonal
direction
u
H
x
• Let’s assume that disturbance propagates a distance L
polewards until polewards pressure gradient balances
Coriolis acceleration (simpler than Taylor’s approach)
• Assuming the relevant velocity is that of the wave, we
R
R
2
(Same as for Rossby l!)
get
L ~
gH  u


F.Nimmo EART164 Spring 11
Baroclinic Eddies
• Important at mid- to high latitudes
Nadiga & Aurnou 2008
F.Nimmo EART164 Spring 11
Baroclinic Instability
Lower
potential
energy
low 
z
high 
cold
warm
• Horizontal temperature gradients have potential
energy associated with them
• The baroclinic instability converts this PE to kinetic
energy associated with baroclinic eddies
• The instability occurs for wavelengths l > lcrit:
l   gH
2
crit
2


Where does this come from?
Does it make any sense?
Not obvious why it is omega and not wave frequency
F.Nimmo EART164 Spring 11
Mixing Length Theory
• We previously calculated the radiative heat flux
through atmospheres
• It would be nice to calculate the convective heat flux
• Doing so properly is difficult, but an approximate
theory (called mixing length theory) works OK
• We start by considering a rising packet of gas:
• If the gas doesn’t cool as
fast as its surroundings, it
will continue to rise
• This leads to convection
F.Nimmo EART164 Spring 11
T
v
• So for convection to
blob
T
occur, the temperature
gradient must be (very
z
slightly) “super-adiabatic”
background
• Note that this means a
(adiabat)
less negative gradient!
z
• The amount of heat per unit volume carried by the blob
 dT
dT 
is given by
z
E  C p T  C p 

 dz ad dz 
• Note the similarity to the Brunt-Vaisala formula
• The heat flux is then given by
 dT
dT 
vz
F  C p Tv  C p 

 dz ad dz 
F.Nimmo EART164 Spring 11
 dT
dT 
vz
F  C p Tv  C p 

 dz ad dz 
• So we need the velocity v and length-scale z
• Mixing-length theory gives approximate answers:
– The length-scale z ~ H, with H the scale height
– The velocity is roughly v ~ H,  is the B-V frequency
• So we end up with:
 dT
F ~ C p 
 dz
dT

dz
ad
 2
 dT
 H  ~ C p 

 dz
dT

dz
ad



3/ 2
1/ 2
g
2
  H
T 
• Does this equation make sense?
• So we can calculate the convective temperature
structure given a heat flux (or vice versa)
F.Nimmo EART164 Spring 11
Key Concepts
•
•
•
•
•
Reynolds number, turbulent vs. laminar flow
Velocity fluctuations, Kolmogorov cascade
Brunt-Vaisala frequency, gravity waves
Rossby waves, Kelvin waves, baroclinic instability
Mixing-length theory, convective heat transport
Re 
uL
n
ul ~(e l)1/3
 NB
g   dT  g 
  

T   dz  C p 
2
l ~ uR / 
1/ 2
 dT
F ~ C p 
 dz
dT

dz
ad



3/ 2
1/ 2
g
 
T 
H2
F.Nimmo EART164 Spring 11