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, M. M., R. Rotunno, 2005: Simulations of Moist Nearly Neutral Flow over a Ridge. J. Atmos. Sci., 62, 1410-1427

Background flow:

2 layer troposphere-stratosphere profile. Moist nearly neutral flow troposphere. Constant wind.

Vertical velocity contours at 5 hrs

Note W cells 100 km upstream of mountain

W perturbation fills depth of troposphere

Associated with W cells is a midlevel zone of desaturated air extending upstream

Cloud water content (white qc < .01 g kg-1).

5 hr

W

qc

Miglietta and Rotunno - investigated saturated, moist nearly neutral flow over topography*Motivation - nearly moist neutral flow soundings observed during Mesoscale Alpine Program. May be important to non-convective flood producing events.

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

Nh/U = .2

W cont .01 ms-1

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 modes1. Linear, hydrostatic analytic solution2. 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

00

1

sin( (1 / )) sin( / ) 1 ( ) 1 ( )( , , )

sin( )t t

n

z Z U n z Z x c t x c tw x z t U

x n n x n x

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

Time-dependent analytic solution

Mountain profile η(x)

Time-dependent analytic solutions for WVary K by changing N and Zt 0-20 hrs

One mode propagating upstream

K (= NZt/ πU0) = 1.15

U=10ms-1, N=.0036s-1, Z=10km

Two modes propagating upstream

W*50 ms-1W*50 ms-1

K (= NZt/ πU0) = 2.3

U=10ms-1, N=.006s-1, Z=12km

Mountain profile η(x)=h0/(1+(x/a)2); h=10m, a=20km

• 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

Wave speed vs K for modes propagating faster than background wind

C- /U

0=

1 -

K/n

Wave speed vs K for c-/U < 0

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

U = 10 m/s, N = .0045/s, Z = 12 km, K = 1.7

time 0 - 5.5 hrs

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)

Nonlinear troposphere-stratosphere

Linear troposphere-stratosphere

Linear – rigid lid replaces tropopause W*50 (ms-1) W*100 (ms-1) W (ms-1)

Nh/U = .68

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

• 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)

U = 20 U = 10

N=.002s-1, Z=11.5km

.37, .73

• 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

WRF saturated simulations- upstream mode 1 speed increases with increasing K

(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 troposphere-stratosphere 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?