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Trapped lee wave drag in
two-layer atmospheres
Miguel A. C. Teixeira1
José L. Argaín2
Pedro M. A. Miranda3
1Department of Meteorology, University of Reading,
Reading, UK
2Department of Physics, University of Algarve,
Faro, Portugal
3Instituto Dom Luiz (IDL), University of Lisbon,
Lisbon, Portugal
2
Trapped lee waves
Energy Flux
dz
dgN
0
2
2/1
2
2
2
2 1
dz
Ud
UU
Nl
Non-hydrostatic
Scorer parameter
Rotors
3
Mountain wave drag
2
0
2
04
lhUD
2D bell shaped ridge: Linear, hydrostatic, non-rotating, constant l limit:
2
0
)/(1 ax
hh
It is known that drag decreases as flow becomes more nonhydrostatic (la
decreases) → this would suggest that trapped lee waves (which are highly
nonhyrsotatic) would produce little drag
However, trapped lee waves exist due to energy trapping in a layer or
interface: wave reflections and resonance → may lead to drag amplification
Aim here is to calculate trapped lee wave drag and untrapped mountain
wave drag in two-layer atmospheres
Important for drag parametrization schemes in global climate and weather
prediction models
4
Linear theory
0ˆ)(ˆ 22
2
2
wkldz
wd
2-layer atmospheres
dkezkwzxw ikx),(ˆ),(
•Boussinesq approximation
•Linearization, 2D flow
•Inviscid, nonrotating, stationary flow
hiUkzw ˆ)0(ˆ
Taylor-Goldstein equation
Waves propagate energy upward or
decay as z w p
Boundary conditions:
continuous at z=H
U
Nl 11
U
Nl 2
2
Scorer (1949) Vosper (2004)
Case 1 Case 2
0
*ˆ)0(ˆIm4)0( dkhzpkdxx
hzpD
Gravity
wave drag p determined from
solutions for w
0
gg
5
Propagating wave drag
Trapped lee wave drag
2 2 22 1 2
1 0 2 2 2 2
1 1 2 10
ˆ| |4
cos ( ) sin ( )
lk h m m
D U dkm m H m m H
Hkn
knkmkhUD
j
jj
j
j )(1
)()(|)(ˆ|4
2
2
2
122
0
2
2
|k|<l2
l2<|k|<l1
Drag normalized by Depends on: 2
01
2
004
hlUD
Hl1 12 / ll al1
Case 1
Propagating wave drag
Trapped lee wave drag
|k|<l2
|k|>l2
2
022
2
22
2
222
01
)(sinh)()sinh()cosh(
))((|ˆ|4
l
dkkHHmkHFrkHkH
kHHmhkUD
Case 2
HknFrFrHknHknHHk
HkHknFrkhkUD
LLLL
LLLL
)()(1)()(
)()()(ˆ
42
22
2
1
2
2
22
2
22
2
2
0
2
2
Drag normalized by 2
02
2
004
hlUD
Depends on: Hl2 Hg
UFr
al2
6 6
0.0 0.5 1.0 1.5 2.0 2.5 3.00.0
0.2
0.4
0.6
0.8
1.0
1.2
l2/(2)
l2/(2)
l2/(2)
jl 2
/(2
)
l1H/
0.0 0.5 1.0 1.5 2.0 2.5 3.0
0.0
0.5
1.0
1.5
2.0
2.5
3.0
NL
l1H/
(a)
Case 1 2.0/ 12 ll
These
quantities do
not depend
on
al1
Numerical simulations 02.001 hl
NL
jl2/(2)
Drag maxima
coincide with
establishement
of new trapped
lee wave
modes
0.0 0.5 1.0 1.5 2.0 2.5 3.00
1
2
3
4
5
6
FLEX
D/D0
D1/D
0
D2/D
0
D/D
0
l1H/
(a)
110l a
0.0 0.5 1.0 1.5 2.0 2.5 3.00
1
2
3
4
5
FLEX
D/D0
D1/D
0
D2/D
0
(b)
D/D
0
l1H/
15l a
21 DDD
7 7
Case 1 2.0/ 12 ll
0.0 0.5 1.0 1.5 2.0 2.5 3.00.0
0.5
1.0
1.5
2.0
2.5
FLEX
D/D0
D1/D
0
D2/D
0
(c)
D/D
0
l1H/
12l a
-10 -5 0 5 10 15 20 25 30 35 40 45 50
x/a
(c)
0
1
2
3
4
5
6
7
8
l 1z/
-10 -5 0 5 10 15 20 25 30 35 40 45 50
x/a
(d)
0
1
2
3
4
5
6
7
8
l 1z/
08.0/ 12 DD5.1/1 Hl
Propagating waves
dominate
7.1/1 Hl 06.0/ 21 DD
Trapped lee waves
dominate
D2/D1 increases as l2a
decreases
w from numerical simulations
1 2l a
10-1
100
101
0
1
2
3
4
5
D1/D
0
D2/D
0
D/D0
FLEX
(b)
D/D
0
Fr
10-1
100
101
0
1
2
3
4
5
D1/D
0
D2/D
0
D/D0
FLEX
(c)
D/D
0
Fr
0.0 0.2 0.4 0.6 0.8 1.00.0
0.2
0.4
0.6
0.8
1.0 l
2H=0.5
l2H=1
l2H=2
Ll 2
/(2)
Fr
Case 2
22l a
jl2/(2) This
quantity
does not
depend on
al2
01.002 hl
5.02 Hl
Numerical simulations
8
21l a
Drag maxima
occur for
Fr 1
D2/D1
increases as
l2a decreases 10
-110
010
10
1
2
3
4
5
D1/D
0
D2/D
0
D/D0
FLEX
(d)
D/D
0
Fr
20.5l a
0 0.4 0.8 1.2 1.6 2 2.4
(l2H)-1
(b)
0
0.2
0.4
0.6
0.8
1
1.2
1.4
Fr
Case 2
D2/D0
Regime diagram of Vosper (2004)
9
• Rotors occur mostly for large j
• There is some correlation between D2 and rotor formation
• This connection must be explored for Case 1 as well
0 0.4 0.8 1.2 1.6 2 2.4
(l2H)-1
(a)
0
0.2
0.4
0.6
0.8
1
1.2
1.4
Fr
jl2/(2)
5.002 hl• Nonlinear ( ) numerical simulations including friction
685.12 al
10
Summary
• Trapped lee wave drag is calculated explicitly for first time for two-layer atmospheres of Scorer (1949) and Vosper (2004).
• Trapped lee wave drag is given by closed analytical expressions.
• Waves trapped within layer may have multiple modes, while waves trapped at temperature inversion may only have single mode.
• Due to resonant amplification, trapped lee wave drag may be comparable to drag associated with waves propagating in stable upper layer, and higher than single-layer, hydrostatic reference value.
• Drag maxima coincide with establishment of trapped lee wave modes (as H increases). For waves trapped at an inversion, drag is maximized for Fr 1.
• D2/D1 increases as l2a decreases. Trapped lee wave drag seems to be maximized for l2a = O(1): wavelength of trapped lee waves matches mountain width.
• Trapped lee wave drag correlates fairly well with occurrence of rotors for Case 2 (still have to check Case 1).
Teixeira, Argain and Miranda (2013a), QJRMS, in press (Early view papers)
Teixeira, Argain and Miranda (2013b), JAS, in press (Early online releases)