Upstream propagating wave modes in
moist and dry flow over topography
Teddie Keller
Rich Rotunno, Matthias Steiner, Bob Sharman
Orographic Precipitation and Climate Change Workshop
NCAR, Boulder, CO 14 Mar 2012
Miglietta and Rotunno - investigated
saturated, moist nearly neutral flow
over topography*
5
W
hr
Motivation - nearly moist neutral
flow soundings observed during
Mesoscale Alpine Program. May be
important to non-convective flood
producing events.
Note W cells
100 km
upstream of
mountain
W perturbation
fills depth of
troposphere
qc
Background flow:
2 layer tropospherestratosphere profile.
Moist nearly neutral
flow troposphere.
Constant wind.
Vertical velocity
contours at 5
hrs
Associated with
W cells is a
midlevel zone of
desaturated air
extending
upstream
Cloud water
content (white
qc < .01 g kg-1).
*Miglietta, M. M., R. Rotunno, 2005: Simulations of Moist Nearly Neutral Flow
over a Ridge. J. Atmos. Sci., 62, 1410-1427
Expanding on Miglietta and Rotunno
• Steiner et al.* conducted a series of 2-D idealized
simulations of both moist and dry flow over
topography
– Similar background flow conditions –
2-layer stability, constant wind
– Varied wind speed, stability, mountain height and half-width
– WRF version 1.3
– Initially focused on comparing long-time solutions for moist and
dry flow
• Investigation of temporal evolution of flow revealed
similar upstream propagating mode as MR2005
*Steiner, M, R. Rotunno, and W. C. Skamarock, 2005: Examining the moisture effects on idealized flow past 2D hills. 11th Conference on
Mesoscale Processes, 24-29 October 2005, Albuquerque, NM.
Example - W and RH for saturated flow
• Vertical velocity (lines)
• Relative humidity (color)
• Animation from 2 to 9 hours
• Desaturated zone associated
with upstream propagating
mode
• Background flow:
Initially saturated
Trop Nm = .002 s-1
U = 10 ms-1
Isothermal stratosphere
• Witch of Agnesi mountain
• height 500 m
• half-width 20 km
RH:
W cont .02 ms-1
Nh/U = .1
But – dry simulations also show
upstream propagating mode
• Vertical velocity contours (color)
• Animation from 3 to 23.5 hours
• Background flow:
U = 10 ms-1
Tropospheric stability .004 s-1
Isothermal stratosphere
• Witch of Agnesi mountain
• height 500 m
• half-width 20 km
W cont .01 ms-1
Nh/U = .2
Upstream propagating wave and
desaturated region in moist flow
•
•
•
Is this related to upstream propagating waves in
dry flow?
Are modes partially trapped by stability jump at
tropopause?
Linear or nonlinear phenomena?
• Use simplified models to investigate
upstream wave modes
1. Linear, hydrostatic analytic solution
2. Nonhydrostatic, nonlinear gravity wave numerical
model
Single layer analytic solution
•
•
•
•
•
Time-dependent, linear analytic solution based on Engevik*
Troposphere only - constant U, N
Rigid lid replaces tropopause
Assume hydrostatic wave motion
Rotunno derived and coded solution for W
*Engevik, L, 1971: On the Flow of Stratified Fluid over a Barrier. J. Engin. Math., 5, 81-88
Time-dependent analytic solution
 sin( (1  z / Zt )) U 0  sin(n z / Zt ) 1 ( x  c t )
1 ( x  c t )
w( x, z, t )  U 0

 n  n   x  n   x 
x
sin( )
 n1
Steady state wave
Left moving
transient
modes
Right moving
transient
modes
• Steady state solution plus sums over left and right moving
transient modes n
• Solution depends on K (= N Zt / πU0 ), i.e., depends on
background wind and stability as well as the layer depth
• Transient wave speed c± = U0( 1 ± K/n)
• Upstream modes traveling faster than the background wind
penetrate upwind (i.e., c- /U0 < 0)
• Number and speed of modes penetrating upwind depends on K
Mountain profile η(x)
Time-dependent analytic solutions for W
Vary K by changing N and Zt
W*50 ms-1
0-20 hrs
W*50 ms-1
One mode propagating upstream
Two modes propagating upstream
K (= NZt/ πU0) = 1.15
K (= NZt/ πU0) = 2.3
U=10ms-1, N=.0036s-1, Z=10km
U=10ms-1, N=.006s-1, Z=12km
Mountain profile η(x)=h0/(1+(x/a)2); h=10m, a=20km
Wave speed vs K for modes propagating
faster than background wind
C- /U0= 1 - K/n
Wave speed vs K for c-/U < 0
• Only transient modes with c- /U0 < 0 actually appear upwind
• Thus for a given K will see only nk modes upstream, where nk is the
largest integer less than K (i.e., nk < K < (nk +1) )
• Speed of a particular mode penetrating upwind depends on K
Numerical simulations – gravity wave
model*
• Use to simulate both rigid lid and linear/nonlinear
2-layer troposphere-stratosphere stability profile
• Time-dependent, nonhydrostatic
• Boussinesq
• Option for either linear or nonlinear advection
terms
• No coordinate transformation – mountain
introduced by specifying w (= Udh/dx) at lower
boundary
• Mountain can be raised slowly
*Sharman, R.D. and Wurtele, M.G., 1983: Ship Waves and Lee Waves. J. Atmos. Sci.,
40, 396-427
Same upstream waves in rigid lid and
troposphere-stratosphere simulations
time 0 - 5.5 hrs
Linear – rigid lid replaces
tropopause
W*50 (ms-1)
Linear tropospherestratosphere W*100 (ms-1)
Nonlinear tropospherestratosphere
W (ms-1)
Nh/U = .68
U = 10 m/s, N = .0045/s, Z = 12 km, K = 1.7
Mountain half-width 20 km height a-b)10 m, c) 1.5 km. W cont. int .05 m s-1, W multiplied by 50 in a), 100 in b)
Upstream propagating waves
• Fundamental feature of both linear and nonlinear
dry numerical simulations
• In both WRF and G.W. models
• Similar to transient modes seen in analytic
solution for single tropospheric layer capped by
rigid lid
• Similar behavior of upstream modes for moist
flow
WRF - upstream modes saturated flow –
vary background wind speed
U = 20
U = 10
N=.002s-1, Z=11.5km
• For stronger background wind speed (U=20 ms-1) all modes
are swept downstream
• As with dry case, 1st mode able to penetrate upwind as K
increases (K10 > K20)
• Similar to dry simulations, except can’t substitute moist
stability in Km (=NmZt/πU0)
.37, .73
WRF saturated simulations- upstream
mode 1 speed increases with increasing K
• W (lines) and
RH (color) at 5 hr
• Speed of wave
and desaturated
region increases
with increasing
Nm (i.e.
increasing K)
• But - can’t
simply use Nm to
calculate K
Nm = .002
Nm = .004
(K=NmZt/πU0; .73 and 1.46)
Saturated background flow
• Transient upstream modes similar to dry flow
• Region of desaturation extends upwind with wave
• What if background flow is subsaturated?
Background flow 70% relative humidity
• W (lines) and RH (color)
• Simulation time 2 hours
• Upstream mode
associated with region of
increased relative humidity
upwind of mountain
• Could transient upstream
propagating wave modes
influence precipitation
upwind of mountain?
Summary • Analytic solution shows transient upstream
propagating waves a feature of linear, hydrostatic
dry flow over topography
• Same modes appear in dry tropospherestratosphere numerical simulations
• Propagation speed depends on tropospheric
wind, stability and tropopause depth
• Speed of upstream propagating wave and
desaturated region in saturated moist flow
follows similar trend
• For subsaturated flow – upstream mode may
increase RH
Are these transient modes important
for orographic precipitation?
• Maybe…
• Numerical simulations contain transients
• Transients can alter moisture content of air
impinging on mountain
• When upstream wave speed only slightly greater
than U0 the transient wave modes may dominate
upwind for hours
• May influence spatial distribution of precipitation
upwind of mountains
• Important to be aware of this possibility when
scrutinizing numerical simulations
• Could play a role when background atmospheric
conditions rapidly changing?