Convective and Mesoscale
Meteorology
MPO 663
Rotation
dominant
diurnal/
PBL
convection
landscape breezes
tides
inertial?
time scale
(2p/f)
latent heating
Precipitation
B-V
period
Buoy,
p*
P evap.
(2p/N)
100 km
Conditional Instability
101 km
102 km
space scale
103 km
Synoptic Instabilities
Rotation
Coriolis
dominant
time scale
diurnal
PBL
convection
landscape breezes
transient
gravity
waves
Stratiform anvils
latent heating
NET
Precipitation
p*
Buoy.
Precipitation
melt / evap
P evap.
destabilization
of sounding
B-V
period
100 km
tides
Conditional Instability
101 km
102 km
space scale
103 km
Synoptic Instabilities
Governing Equations
• Boussinesq for clarity:  is constant (like
water), and hydrostatically balanced by
basic-state pressure p(z)
• can relax to anelastic (z) for air
• The residual p’ = p - p drives motion
• Bundle as  = p’/ (= p in earlier notes)
• Thermo is in terms of buoyancy b
» g’/= gv’/v= g’/ for dry air
• Dynamical equation set has u,v,w,b,
• 5 eqs: momentum(3), 1st Law, Mass cons.
• ignore vertical Coriolis terms
Boussinesq equations
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•
u
v
Mass
w
Thermo
With driving mainly by heatings:
» latent
radiative surface/turb etc.
Parcel theory: just thermo equation
• Entropy of parcel = S(Tinit, qinit, pinit)
• Sp conserved, as is qtotal, EXCEPT
• mixing with some “environment”
• fallout of condensed water
• freezing/melting/sublimation
•
•
•
•
T(z) from invert_S(Sp, qtotal, p)
Tv_env(z) from some sounding
b(z) = g [T(z)-T(z)]/T(z)
implied relevance to vertical accel.
» dw/dt = b (-z if fussy) (= wwz in steady case)
Gravity waves: neglect Coriolis, advxn, F, Q
(weak, unforced motions; faster than 1/f)
• u
• v
• Mass
• w
for non-hydrostatic
case (cosnot)
• Thermo
• Take all  ei(kx+ly+mz-t)
z
• ”dispersion relation”
 =
2
N (k 
2
(k 
2
2
2
2
m )
)
2

= N cos
x
Gravity waves propagating right to left
L
H
L
H
L
H
H
L


H
L
H

L
pressure () accelerates the
horizontal wind as needed
A wave packet: a finite band of Fourier
components, superposed and interfering

H


H
L
Group velocity
(packet energy
propagation)
H
L
L
Phase velocity
Boundaries are crucial
• Pure ei(kx+ly+mz-t) solutions are impossible
(unbounded oscillations across space and time)
• impose boundary conditions
• like w=0 at z=0
result: reflected waves
w = 0 everywhere along this line
Or, the boundary condition can be a
wave source: flow over hills
(or “push a corrugated sheet under the atmosphere”)
Is there a “lid” up there?
• Scorer parameter (for waves in sheared
flow over mountain) is a generalized N2
•
http://moe.met.fsu.edu/~rhart/mtnwave.html
– Matlab tool
Flow over
mountain
(or mountain
“dragged to
the left” if you
prefer)
more stable:
for same
frequency,
 is larger
(steeper tilt)
less stable
Trapping of waves
• stable layer, under smaller l (Scorer 1949)
• double reflection => “vertical mode” of cavity
– can propagate long distances horizontally
example
http://commons.wikimedia.org/wiki/Image:Wave_cloud.jpg
The other main source: Q
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•
•
u
v
Mass
w
Thermo
• Nicholls, M. E., R. A. Pielke, and W. R. Cotton, 1991: Thermally
forced gravity waves in an atmosphere at rest. J. Atmos. Sci., 48,
1869–1884.
• Pandya, R. E., and D. R. Durran, 1996: The influence of
convectively generated thermal forcing on the mesoscale circulation
around squall lines. J. Atmos. Sci., 53, 2924–2951. Also 2000
extension to 3D
w field emphasizes vertical
propagation
here for a transient heating event (Holton book CD)
b,u fields show horiz. propagation
Animation of solution with lid
MCS-like heating in resting atm.
Times just after a period of deep heating has ended
0-50
m/s
-50
m/s
-50
m/s
m/s
Warm
Warm
Warm
Warm
Warm
5050
m/s
50
m/s
50
m/s
m/s
Warm
Warm
Warm
Warm
Warm
propagation speed
c = N/m = (vertical wavelength)/(~10 minutes)
More realistic
(Q produced by time
averaging a squall
line in a cloud
model)
Conclusion:
Much of squall-line
mesoscale
circulation can be
viewed as a gravity
wave field forced by
latent heating
Pandya and Durran
1996
vague
arbitrary
figure
shown for
no clear
reason
Atm has no lid - are vertical “modes” real?
full-physics squall line
in a cloud model with
strong vertical shear
& rad. upper BC
Pandya and Durran 1995
“2” stays vertical
despite advection that
would tilt it severely!
(trajectories below)
Very hard to observe
• Deep heating driven gravity waves
– 1/2 wavelength in troposphere: ~50 m/s
– 1 wavelength in troposphere: ~25 m/s
– 3/2 wave: ~15 m/s
• spread heating over a large area quickly
– so dT/dt is small
• wind signal is divergent (ageostrophic)
– hard to observe under all the rotational wind
• Shallower vertical structures are easier
Convective outflows (and other
things) also drive trapped “undular
bores” on low-level stable layers
Add the Coriolis force: inertio-gravity waves
• u
• v
Phase velocity
Group velocity
 
2
f 2 m2  N 2 (k 2 
(k 2 
2
2
)
 m2 )
• but f << N, so this term only matters
when m2 >> (k2 + l2), i.e. for broad
scale waves (hydrostatic). But then
the denominator is just ~m2 so
z
Phase front
wind vector
rotates
elliptically
w/time
• still c = N/m
Classic heating-driven inertiogravity wave: sea breeze at 30o
Simplest low freq. limit: geostrophic vortex
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•
•
•
•
•
•
•
u
v
Mass
w
Thermo
Driven by previous heating (HW3)
Defined by its PV
vtan  e(-x/R) where R = c/f = N/mf
= [(vert. wavelength)/(10’)]/f is Rossby radius
Geostrophic vortex: heat is trapped,
waves can’t carry it beyond c/f
W
W
Rossby radius
C
W
W
tangential winds in geostrophic (x-z)
vortices after 2- mode heating
MCV
Rotation
dominant
time scale
diurnal
PBL
convection
landscape breezes
cascades
transient
gravity
waves
Stratiform anvils
Net latent heat
Precipitation
Precipitation
melt / evap
p*
Buoy.
tides
P evap.
B-V
period
100 km
Conditional Instability
101 km
102 km
space scale
103 km
Synoptic Instabilities
Rotation
dominant
TCs
diurnal
PBL
convection
landscape breezes
time scale
fronts
tides
MCCs
MCVs
transient
gravity
waves
Stratiform anvils
rainbands
Net latent heat
Precipitation
supercells melt / evap
Precipitation
p*
Buoy.
P evap.
B-V
period
100 km
Conditional Instability
101 km
102 km
space scale
103 km
Synoptic Instabilities
When advection is dominant...
•
•
•
•
•
u
v
Mass
w
Thermo
0
0
Turbulence: vortices advecting vorticity
• A vortex filament or tube “induces” flow
– tangential velocity v falls off as 1/r
– circulation G = 2pr v = const for any circle enclosing the filament
» wind everywhere from a localized vortex tube
» wind is like “flesh” on the “skeleton” of vorticity
http://en.wikipedia.org/wiki/Image:Vortex_filament_%28Biot-Savart_law_illustration%29.png
Supercell
scale: vortices
have dynamic
pressure
1-10ish km scales
MCVs 10-100ish km
MCVs
not so much dynamic pressure at
this scale, but same tilting term can
makes a hor. circulation that can
advect things (bookend vortices)
Scale interactions: triads
• Advection terms, expanded as products of Fourier
series, contain sum/difference wavenumbers
• p, q, p+q form a triad of interacting wavenumbers
• Triads may be local in scale (same octave)
• or nonlocal
p=15, q=1
p+q = 16
-------->
energy 
“Scale” vs. wavenumber in spectra
scale = log wavelength 
• Scale is fundamentally logarithmic
– octaves (factors of 2) or decades (factors of 10)
• 1km, 2km, 4km, …
or 1’s, 10’s, 100’s,… of km
– small scales have more wavenumbers per scale
– so, for example, white noise is dominated by small “scales”
» careful with fft spectra and plotting conventions!
3D isotropic turbulence
• Homogeneous (not stratified)
• Spaghetti of vortex filaments
• A classic physics problem
• simple to specify, hard to “solve”
3D turbulence: energy source is at large scales
waves and instabilities
solid
boundary
conditions
buoyancy sources
3D turbulence: sink is at small scales
Big whirls have little whirls,
which feed on their velocity.
Little whirls have lesser whirls,
and so on to viscosity.
L.F. Richardson
1925
If the transfer
occurs
between
eddies of
similar
sizes, it is
said to be
a cascade.
Nonlocal in scale: weak interaction
+
+
Cascade consequences
• Slope of spectrum MUST BE -5/3
– from purely dimensional considerations
• Energy source (large scale) = sink (small scale)
• Between is conservative (inertial) cascade
– powerful! doesn’t depend on mech of scale interaction
2D turbulence: upscale TKE cascade
from small-scale sources (convection?)
to large-scale sink (F ~ -eu)
When little whirls meet little whirls,
they show a strong affection;
elope, or form a bigger whirl,
and so on by advection.
http://www.fluid.tue.nl/WDY/vort/interact/img/dipcol4s.gif
http://www.mae.cornell.edu/fdrl/research/vortex_interactions.html
http://www.student.math.uwaterloo.ca/~amat361/Fluid%20Mechanics/topics/vorticity.htm
Local
in scale
vortex
interactions
(vortices are of
comparable
size)
Prieto et al 2003 JAS
vorticity
Energy gets
coarser,
enstrophy
finer
Vallis 2006 textbook
streamfunction
time
3D vs. 2D cascades: both -5/3
3D downscale of forcing, 2D upscale of it
3D
large scale
damping
-u/t
2D
small scale
source
(convection?)
vorticity fields with dangerous
(to predictability)
vs. feeble scale interactions
SQG system
Limited Predictability
barotropic system
Unlimited Predictability
Rotunno and Snyder 2008
Convectively coupled gravity waves
(2D CRM, Tulich)
5 decades
}
3 decades
Can we conclude it’s a cascade?
Not with rigor.
Energy budget signature of a
‘classical’ 2D-Hor cascade:
KE budget with scale, cumulative view
convection drives small scales
[w’b’]
cumulative
KE source
(norm.)
damping at
inertial range
has a
advection
conservative
cascade
advection
sends energy upscale
large
scales
(-u/t)
Is 2D CRM energy spectrum like
a cascade?
buoyancydriven KE
source at small
scales
inertial term
removes KE at
<10km scales
and deposits it
in 10-50 km
range
convection
source active
throughout 5010,000 km
range
no residual
advection
a dynamical
mesoscale
a dynamical
large scale
Stefan Tulich
advection (inertia)
also collects
energy from 5010,000 km
wavelengths and
deposits it in
“planetary”
wavenumbers 1-4
Convectively coupled gravity waves
(2D CRM)
5 decades
}
3 decades
What can we actually conclude? Nothing with rigor.
‘Characteristic’ scale depends where you start
5’ point cloud&rain
6h’ly q&rain
4d means
}{
{
}
{
}
}
e-1
daily means
6h’ly
conv
}
}}
e-2 meso
Mapes et al. 2006 DAO
Energy budget signature of a
‘classical’ 2D-Hor cascade:
KE budget with scale, cumulative view
convection drives small scales
[w’b’]
cumulative
KE source
(norm.)
damping at
inertial range
has a
advection
conservative cascade
advection
sends energy upscale
large scales
(-u/t)
Is 2D CRM energy spectrum like
a cascade?
buoyancy-driven
KE source at
small scales
convection
source active
throughout 5010,000 km range
no residual
inertial term
removes KE at
<10km scales and
deposits it in 1050 km range
advection
a dynamical
mesoscale
a dynamical
large scale
Stefan Tulich
advection (inertia)
also collects energy
from 50-10,000 km
wavelengths and
deposits it in
“planetary”
wavenumbers 1-4
Predictability: revisited by Rotunno and Snyder 2008 JAS
SQG*
2D
LimitedSQG
Predictability
Unlimited Predictability
2DV
E (k ) ~ k 5/ 3
E (k ) ~ k 3
up-scale
error growth
from rapidly sat’d
small scale
errors
up-amplitude
error growth
at each
scale
t 0
E (k , t )
1
k
104 1
*SQG=Surface QG , Blumen(JAS 1982)
Pierrehumbert, Held and Swanson (CSF 1994)
E (k , t ) t  0
4
k
10
104
Spectral view of predictability issue:
NICAM 12km global model wind
Error growth
appears to go
up-amplitude
not
up-scale
potentially good news
for long predictability
Mapes et al. 2008
eff.
res.
Spectral view of predictability issue:
2. 2D CRM layer-mean KE
Again,
errors grow
largely
up-amplitude,
not
up-scale
Mapes et al. 2008 JMSJ
un-log that
y (power) axis:
a)
30d
16d
(Lorenz’s diagram, backward)
• difference growth in 2
realizations of NICAM runpairs with near-identical
initial conditions
•
2 4
8d
30d
b)
... but recall complication: runs differ
in resolution (7km vs. 14km mesh),
not just tiny initial condition diffs...
Mapes et al. 2008 JMSJ
16d
1
4
8d
2
Difference Energy of Band-Pass Filtered Fields
E
S  [
  200km]
M  [200    1000km]
L  [
  1000km]
t[ h ] 
Zhang et al. (JAS 2007)
• Marat
Khairoutdinov
(Stony Brook)
ran “Giga-LES”
• Moeng et al.
2009, 2010
JAMES
Split the LES flow into: “resolvable” grid-scale (GS)
& “unresolved” scale (SGS)
Giga-LES
apply
“smoothing”
CRM resolvable
SGS is the difference.
Moeng et al. 2010 JAMES
Apply “smoothing” with a width of 4 km
GS
SGS(w-var)
SGS(q-var)
GS
GS
large scales
SGS (wq-cov)
GS: CRM-grid scales
SGS: CRM-SGS
most of w-kinetic energy in SGS
~ half of moisture flux in SGS
small scales
Moeng et al. 2010 JAMES
• SGS flux
Moeng et al. 2010 JAMES
• SGS flux is
in clouds
• condensedw
ater path
(vertical
integral)
Moeng et al. 2010 JAMES
Flux partitoned by scales
• Vapor flux by sub-convective (0.1-4km) scales
is colocated with >4km scale convective cloud
updrafts.
• Small scales mainly just add a bit (~40%) to the
flux by convective mean updrafts
• Vapor flux by convective (5-80km) scales is
colocated with flux in >80km scale mesoscale
cloud system updrafts.
• Small scales mainly just add a bit (up to 40%)
to the flux by mesoscale mean updrafts
Latent flux
across
500mb
snapshot
by scales
resolved
in 80km
rebinning
sub80km =
total flux
sub-80km and super-80km scales
conspire to carry flux: convection
occurs in mesoscale clusters
Flux part summary
•
•
•
•
Mesoscale updrafts are moist, fluxing q up
Convective updrafts are inside, adding to it
Sub-drafts inside the convective drafts: ditto
Q: How might poorly-resolved convection be
distorted by having to carry the flux of missing
sub-scales? (and can param’z’n fix it?)
• Q2: Is subgrid param’z’n a flux amplifier? Is
that safe numerically?
Advection destroys balanced flow
“Secondary” circulations driven by
rotational (geopstrophic) flow
QG Omega Equation in Q-vector form:

 2 f0  
 p 
2     2  Q
 P 

2


Q  Qx ,Qy

2



Vg
RP 1  Vg

 
  p ,
  p 
 
y
P0  x

PV view of how maintenance of balance
in the presence of shears that would destroy it
implies vertical motions
Advective Complications
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u
v
Mass
w
Thermo
• Latent