phase and group velocity
DESCRIPTION
phase and group velocity. Note: NOT vector components! (“trace speeds”) . Simplest case: Ignoring f, why not align x axis with waves (no equation 2a). Form a vorticity equation from 1a, 3a (eliminating p’) Use mass continuity (4a) to eliminate u. (w xx + w zz ) tt = N 2 w xx - PowerPoint PPT PresentationTRANSCRIPT
phase and group velocity
Note: NOT vector components! (“trace speeds”)
Simplest case: Ignoring f, why not align x axis with waves (no equation 2a). Form a vorticity equation from 1a, 3a (eliminating p’)Use mass continuity (4a) to eliminate u.
(wxx + wzz)tt = N2 wxx
Assume wavelike solutions exp(i (kx+mz-st)) w2 = N2 k2 (k2 + m2)-2
w = N cos q
• freq = N cosq• (slope of parcel
motion paths)• http://www.atmo
s.ucla.edu/~fovell/DTDM/
• http://www.atmos.ucla.edu/~fovell/AOS101/sgw.html
Note: momentum is being fluxed (u’w’)
Mountain waves
Courtesy: Geraint Vaughan
Inertio-gravity waves• These rather forbidding equations hide a pleasing elegance to the
gravity-wave solutions which become more evident if x is taken to be the direction of propagation of the wave. Then ℓ=0 and V = -iUf/ω. V and U are in quadrature, causing the wind vector to rotate elliptically – clockwise for upward energy propagation and anticlockwise for downward. For oscillations near to the inertial frequency (which are common in the lower stratosphere) the ellipses are nearly circles, as shown in the example below measured by a VHF radar at Aberystwyth, Wales.
Geraint Vaughan
Taylor-Goldstein eq• Don’t assume wavelike in z, only in x & t• Permit U(z), and r(z) (Anelastic)
Taylor-Goldstein eq• Don’t assume wavelike in z, only in x & t• Permit U(z), and r(z) (Anelastic)
Critical level, where u = cor intrinsic frequency 0
• dumps westerly momentum below critical level– think QBO (downward propagation of u)
“overreflection”: ideal for wave
trapping/ducting
“Importance” of gravity waves1. For middle atmopshere: Systematic upward
flux of zonal momentum [u’w’]– goes as (wind amplitude) x (tilt of motions)
• tilt goes with frequency (w = N cosf) – high frequencies are highly weighted in this importance
metric
– systematic E/W asymmetry• asymmetric sources
– form drag in flow over obstacles (mtns or cloud tops) • asymmetric filtering on the way up (critical levels etc.)
– even w/ symmetric sources, but base state flows/ shears
– detectability an issue for global assessment• vertical wavelengths for satellite kernels, etc.
“Importance” of gravity waves1. Notes: w’T’ = 0 so they don’t carry heat flux
vertically 2. often viewed in terms of w field which
emphasizes short wavelength and high frequency waves
3. These ideas can almost close off from view another important role for long (hydrostatic) waves: as an adjustment mechanism
1. carrying heat away from convection horizontally2. adjusting stratification profile
1. to moist adiabatic
“Importance” of gravity waves 2.
1. adjusting stratification: vertical displacement– strong interactions with convection– goes as (w amplitude) x (period or duration)
• low frequency components especially important– like from net heating events (zero frequency limit)
– a whole different slice of (k,l,m,w) space• which is only really 3D since dispersion relation holds:
M&R Bores: horizontally moving wavefronts• Parsons
ppt on 663 course web page
Back to simple waves, in resting Boussinesq fluid.Hydrostatic, or low frequency limit
(small k & l, i.e. m >> k,l)
Aligning x with the wave (l=0), w = N k/mc = cg = N/m = vert. wavelength/ BV period“Horizontally propagating vertical modes”
Vertical modes
• w=0 BC at top and bottom of stratified layer discretizes allowable vertical wavenumbers m– w = sin(kz) – wave numbers ½, 1, 1.5, ... of the layer– waveguide or duct – no vertical propagation, or upward + downward propagation
are equal (reflections) yielding standing oscillations (vertical modes)
– The higher you place the lid, the closer together the wavelengths permitted
Example: response to deep convective heat sources
• Boussinesq, hydrostatic, constant-N, nonrotating, resting basic state– w = N k/m is dispersion relation for waves
• Heating is forcing, confined to ‘troposphere’– lower portion of deep stratified fluid under lid
• Heating turns on at t=0 and then maintained • 2 modes of tropospheric layer
– convective (sin(z) all positive)– stratiform (sin(2z) vertical dipole)
w solution 20 hours later• radiating
upward• slope of ray
path0 set by 20h timescale since heating began
acts like virtual dipole source in infinte fluid since BC is a symmetry line (w=0)
Mapes JMSJ 1998
Lids cause errors: interference w/artificially reflected waves
• Here steep rays are those set by k,m
• Their arrival peapods w solution
Mapes JMSJ 1998
displacement (T’) field is far less wrong than w’ field, even for 10km lid! Vertical propagation and reflection error misses the point.
Vertically propagating waves don’t carry heat upward
• but they did carry it outward
Mapes JMSJ 1998
Let’s just look at tropopase lid case
• here for f=0, 6h heating event 16 hours ago
w field at t=16h, 10h after a 6h heating event happened at the origin.
T field for same case
What does f do? • Traps heat within a Rossby deformation radius
of the origin, held back by thermal wind shear
inertio-gravity waves
b trapped
inertio-gravity waves (zero-sum waves) escape
Solution procedure
€
˙ Φ n (x,t)∝ Q(x,z, t)sin(2πz /Lzn )dz0
Z top
∫ ; Lzn = 2ztop /n
Fourier transform in zalong with B.C.s
Unknowns: u(x,z,t), w(x,z,t), T(x,z,t), p(x,z,t)
Known: Q(x,z,t)
Shallow water system (1 for each Fourier mode: n = 1, 2,3,..)
Since N is const.
Governing equations
€
∂un∂t +∂φn∂x=0 (1)
€
∂φn∂t +cn2∂un∂x=˙ Φ n (2)
Stefan Tulich ppt
Sketch of solution*
€
w ~ ±sin(πz /D) → Lzn = 2D
Vertical velocity w at times ti < t < tf
cn-cn
€
cn = ± LznN 2π = ± DN π ≅ 50m s−1
*References: Nicholls et al. 1991; Mapes 1998
Stefan Tulich ppt
Sketch of solution
Horizontal velocity u at time t < tf
50 m/s-50 m/s
Stefan Tulich ppt
Sketch of solution
Temperature T at time t < tf
50 m/s-50 m/s Warm
Stefan Tulich ppt
Sketch of solution
Times after the heating is switched off (t > tf)
WarmWarm 50 m/s-50 m/s
Stefan Tulich ppt
Sketch of solution
Times after the heating is switched off (t > tf)
WarmWarm-50 m/sWarm 50 m/sWarm-50 m/s
Warm-50 m/s
Warm 50 m/sWarm 50 m/sWarm 50 m/sWarm-50 m/sWarm
“Modes”? Convective and StratiformExample: 2 radar echo (rain) maps (w/ VAD circles)
200 km
Convective & stratiform “modes”
Con
Con
Strat
Strat
In pure simplest theory case
Con: sin(z)
Strat: sin(2z)
Houze 1997 BAMS
Addition of a stratiform-like source
Spatial structure:
cooling
warming
€
Qst ∝−sin(2πz /D) → Lzn = D
tDCi tDCf = tStitime
QDC0 QST0
tStf
Temporal structure:
Stefan Tulich ppt
Response to stratiform heating
Just before the heating is switched off
50 m/swarmwarm
cold
25 m/s
Stefan Tulich ppt
Longer vertical wavelengths travel faster horizontally
A complex convective event in a salt-stratified tank excites many vertical wavelengths in the surrounding fluid (photo inverted to resemble a cloud).
Strobe-illuminated dye lines are displaced horizontally, initially in smooth, then more sharply with time.
Mapes 1993 JAS
early
late
Glimpses of invisible env. flow
Fourier spectra of QDC and QSt for different lid heights
Ztop = DZtop = 2DZtop = 4DZtop ∞
€
˙ Φ n (x, t)∝ Q(x,z,t)sin(2πz /Lzn )dz0
Z top
∫ ; Lzn = 2ztop /n
positive coeff. implies warmingresponse near the surface
negative coeff. implies cooling response near the surface
Lidded solutions are crude approximations to continuous solution
Multiple wave packets are excited
Stefan Tulich ppt
Summary of wave/mode background
• The flow of stratified clear air outside convective clouds is dispersive– longer vertical wavelength components
expand faster/farther away from source horizontally
• Any vertical profile, e. g. divergence, can be expressed as a spectrum, w/ axis labeled by phase speed. – lid discretizes spectrum; bands robust
Is all this sin(z) ghost/mode stuff realistic? (or kinda kooky?)
• Need: modes of a realistic atmosphere (actual r stratification profiles)–Ready: Fulton and Schubert 1985
• Need: realistic heating (divergence) profiles–Ready: many many VAD measurements
What about for real atmospheres where N is not constant?
We can still use the same procedure but a more general (“vertical mode”) transform must be used:
€
˙ Φ n (x, t)∝ f [Q(x,z,t)]ψ n (z)ρdz0
Z top
∫
The vertical structure functions n (and their associated phase speeds cn) are obtained as numerical solutions to the vertical structure problem: an eigenvalue problem with dependence on N
Stefan Tulich ppt
Vertical modes associated with the CRM’s basic state atmosphere*
*calculated using the algorithm of Fulton and Schubert (1985)
Lzn ≈ 28 kmLzn ≈ 14 km“ Lzn” ≈ 11 km“
cos(2z/Lzn)
Structure functions of both u and f
Spectrum of average VAD divergencefrom many profiles in tropical rain
different lid pressures ->
different discretizations,
bands robustHey -- what’s this?
Mapes 1998
T response when
observed mean VAD
divergence is used as a
mass source in observed
mean stratification
Shallow water solved for
each mode, then sum it up
Mapes and Houze 1995
Top-heavy C+S: spectrum & response
Melting: forcing is
localized in z, response is localized in
wavenumber!
Melting mode
Mapes and Houze 1995
Raw data: Snow melts,
whole troposphere
shivers
(wavelength set by melting layer
thickness?)spectral view not quite so
kooky?
m=1m=3/2
Does this exist?
m=1/2
Re: kookinessAre convective and stratiform really dynamical modes?
Rare, but compelling
(great data quality)
Jialin Lin
Rare, but compelling (5h of data, from front to back of
storm)
Aboard the R/V BrownJASMINE project
considerable front-back
cancellation
May 22, 1999(figs from U. of Washington webpages on JASMINE)
~15 m/s
Webster et al. 2003, Zuidema 2003
In a storm notable for fast, long-distance propagation
diurnal
Kousky - Janowiak - Joyce (NOAA CPC)
ship
Re: kookinessnumerical modeling, with advection and no lid
Pandya and Durran 1996
u
u later
Re: kookiness
Wavefront 2 stays vertical and coherent despite advection by sheared winds nearly half the wave speed!
Pandya and Durran 1996
Re: kookinessmore numerical modeling
Even convective cells appear to be gravity waves!?
Yang and Houze 1995
This stuff hasn't totally sunk in to the convection community (myself
included!)
Spectral questions
• Where do the observed modes come from ultimately?
Modal (band) responses seen away
from convection
• Yes, Convective and stratiform “modes” seen in T fluctuations, but
• ~15 m/s also prominent
Fast ghosts zipping everywhere - only
statistics are available reliably
?
A fundamental source for c ~ 15 m/s
radiativecooling12km
moist adiabat runs dry
8km
spectrum of square Qrad forcingobs.
strat.
NO fundamental source for c ~ 25 m/s ("stratiform mode")
• Apparently excited by processes internal to convective cloudiness
– half-troposphere depth cumulus congestus rainclouds
– precipitating stratiform anvil clouds
No fundamental source -> GCMs failLack of stratiform processes, or of cumulus showers?
GCM
Deep convectionheating in GCM
Lee Kang Mapes 2001
20N-20S cooling
Deep convectionheating
obs
Earth
Mapes 2000
Cloud resolving model has it...
Tulich Randall Mapes 2006
shallow cu (SC) & stratiform (ST)
opposed
SC only in lower half of
mode
Revisiting the outline• (Intro: white lumps, invisible
environs)• Spectral laws of stratified flow• “Modes” of convection • The life cycle: why grow just to
die?
– A question of coupling between the 2 halves of convective circulations
»(the white part (clouds) + spectral (dispersive by m) environmental flow)
Stratiform instability• Recall d(KE + PE) = [b’Q’] • Heating where it’s hot drives all disturbance energy• But buoyancy driven convection should favor cold
places.• Low-level b key
– Inhibition control
http://journals.ametsoc.org/doi/full/10.1175/1520-0469 (Mapes 2000)
Occurs on many scalesbigger<-> longer lived, suggests a velocity
scale
Mapes Tulich Lin Zuidema 2006
Clean: 4000 km rain waves in a 2D model(All the following
work by Stefan Tulich)cc3
The life and death
of cc3
a multicellular
entity
shallow
deep
strat.
Why die? Why do new cells fail?
1 km warm T’
BUOYANCY OF LIFTED AIR PARCELSFROM LOW LEVELS
env warm
(& dried)
cell-killing warm wedge: a downward displacement in a wave
warm T’ cold pools slide
under, but new cu fail
What does the LS wave look
like?
a larger version of
cc3, of course!
cu in front
deep
strat.
LS wave motion to right
Note T’ no bigger in heated areas -
equilibrated wave
Front edge: wave forces cu clouds
cu heating nestled in low T’, which keeps falling
But why does the large scale wave exist?Must go back to origins
(different model run - main wave went R->L)
widening riverof wave
amplitudeas events trigger
next events
Key mechanism: short vertical wavelength mode
change it via radiative cooling
depth and/or lapse rate
changed wavelength spectrum
rain wave speed
changes accordingly
Conclusions• Illusion of clouds as substantial is
visually compelling–Must be resisted with rationality
• Motions of embedding environment are inseparable, and spectral–Longer vert. waves travel faster –chromatography of outgoing signals–sloped destabilization by incoming signals
Not kooky, but a little spooky• Artifice of upper lid not too bad–believe bands not modes
• (but mode is a convenient word)• Neglect of advection not too bad–wavefronts remain upright & coherent
even in strong shear• how ??
–secondary circs?
Where does wave-1 of troposphere “mode” come from?
• Precipitating stratiform anvils force it• Cumulus congestus showers force it
» lower half only
• These cancel on average - there is no physically fundamental source
»large-scale models can miss it via parameterization errors
Convective & stratiform
–Inevitable microphysical outcomes of bubble ascent (rain, ice, etc)?
–Or dynamical modes of motion?•What governs downdraft depth for example?
»rain could just saturate air & stop evaporating if descent didn’t agree with the ambient airflow...
Leading edge of the life cycle• Is this 2000 km / 20 hour wedge scale
governed by the cumulus dynamics of moisture buildup?
• Or does wave cooling invite (by buoyancy) or demand (for balance) a certain heating?
» Sensitivity to precipitation efficiency of cu?
shallow cu heating
Is the MCS just another convectively coupled wave type?
• small scale, large amp., but qualitatatively...
What’s up with this?
Substantial, very repeatable
deviation from a moist adiabat.
CRMs don’t get it.microphys (e.g.
ice?)small cu effects?
LS (trades) crucial?
Discussion welcomedmapes @ miami.edu
Thank you!