Department of Meteorology

Trapped lee waves: a currently neglected source of low-level orographic drag

Miguel A. C. Teixeira1, Jose L. Argain2, 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

www.met.reading.ac www.reading.ac. uk k Department of Meteorology Trapped lee waves Non-hydrostatic

dg θ N 2 = Energy Flux θ0 dz

Scorer parameter 2/1  N 2 1 2Ud   −=  l  2 2   UU dz 

Workshop on drag processes – 12-15 2 Sb 2016 Department of Meteorology Mountain wave drag

Important for drag parametrization schemes in global climate and weather prediction models

It is known that drag decreases as flow becomes more nonhydrostatic (narrower obstacles) → this would suggest that trapped lee waves (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

How is drag partitioned into trapped lee waves and vertically propagating (untrapped) mountain waves?

Bell-shaped 2D and 3D Linear, hydrostatic, non-rotating, circular mountains constant l limit

h0 h0 π h = h = 2 2 π 2 2 2 2 2 2/3 0 = ρ0UD lh0 = ρ lahUD + ax )/(1 [ ++ ayax )/()/(1 ] 4 0 4 0 0

Workshop on drag processes – 12-15 3 Sb 2016 Department of Meteorology Linear theory

+∞+∞ •Linearization, Boussinesq approximation + ),,( = ˆ ),,( 21 ykxki )( dkdkezkkwzyxw •Inviscid, nonrotating, stationary, uniform flow ∫ ∫ 21 21 ∞− ∞− 2wd ˆ 2 + kk 2 + 1 2 2 2 ˆ =− 2 2 1 )( wkl 0 Taylor-Goldstein equation dz k1 ˆ Boundary conditions: ˆ zw )0( == iUk1h

wˆ pˆ continuous at z=H Waves propagate energy upward or decay as z→∞

2-layer atmospheres Scorer (1949) Vosper (2004)

N l = 1 Case 1 Case 2 1 ∆θ U ∆θ ′ = gg θ0 N l = 2 2 U

Workshop on drag processes – 12-15 4 Sb 2016 Department of Meteorology Gravity wave drag +∞+∞ ∂h +∞+∞  pˆ determined from zpD == )0( dxdy = π 2 Im8 ˆ = )0( ˆ* dkdkhzpk ∫ ∫  ∫ ∫ 1 21  solutions for wˆ ∞− ∞− ∂x  ∞− 0  Case 1 Propagating wave drag (2D) l2 k|| hˆ 22 mm DU= 4πρ 2 12 dk |k|

Trapped lee wave drag (2D) 2 knkm )()( = 2 ρπ 2 ˆ 2 1 2 jj D2 4 0 ∑| khU j |)( l2<|k|

1 km j )( Resonance condition (2D) []1 j )(tan Hkm −= 2 kn j )(

π π Depends on = ρ 2 2 2 2 Drag normalized by 0 0 hlUD 01 or 0 = ρ0 ahlUD 01 4 4 1Hl / ll 12 1al Workshop on drag processes – 12-15 5 Sb 2016 Department of Meteorology Case 2

Propagating wave drag (2D)

l2 ˆ 22 2 | (| 2 )(kHHmhk ) 1 = 4πρ0UD dk |k|

Trapped lee wave drag (2D)

2 2 ˆ −2 2 2 khk LL )( {[Fr − 2 L ] − L HkHkn )()( } = 2 ρπ 2 D2 4 0U 2 −1 −− 22 |k|>l2 L [][2 L +++ 2 L )(1)()( HknHknHHk − ][FrFr − 2 L )( Hkn ]

6 Hk = L Resonance condition (2D) tanh()L Hk −2 Fr − 2 L )( Hkn

Depends on π π = ρ 2 2 2 2 Drag normalized by 0 0 hlUD 02 or 0 = ρ0 ahlUD 02 U Hl al 4 4 Fr = 2 2 ′Hg

Workshop on drag processes – 12-15 6 Sb 2016 Department of Meteorology Case 1 (2D): Drag Results for ll = 2.0/ 12 = 6 Numerical simulations hl 01 .0 02 (a) FLEX 5 5 D/D (b) 0 FLEX D /D 1 0 D/D 4 4 0 D2/D0 D1/D0

D2/D0 la= 10 0 3 = 1

D la 5 3

/ 1 D 0

D 2 / la2 = 2 D 2 la2 = 1 1 1 0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 0 l H/π 0.0 0.5 1.0 1.5 2.0 2.5 3.0 1 2.5 l H/π (c) 1 FLEX D/D 2.0 0 D /D 1 0 • D2/D0 may be large (~3) D2/D0 1.5 • Drag maxima coincide with la= 2 0

1 D / establishment of trapped lee wave D 1.0 modes 2al = 4.0 0.5 • Agreement with numerical simulations requires considering both D and D 0.0 1 2 0.0 0.5 1.0 1.5 2.0 2.5 3.0 π • D /D increases as l a decreases l1H/ 2 1 2

Workshop on drag processes – 12-15 7 Sb 2016 Department of Meteorology Case 1 (3D): Drag

3.6 Numerical 3.0 3.0 Total drag (theory) Numerical Internal (theory) Total drag (theory) Lee wave (theory) 2.5 Internal (theory) 2.4 Lee wave (theory) la1 = 10 2.0 0 1.8 la= 5

1 0

D/D 1.5

D/D la2 = 2 1.2 la2 = 1 1.0 0.6 0.5 0.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 0.0 π 0.0 0.5 1.0 1.5 2.0 2.5 3.0 l1H/ π l1H/

1.6 Numerical → 1.4 Total drag (theory) • D2/D0 may be large (~2) some Internal (theory) 1.2 Lee wave (theory) directional wave dispersion la= 2 1.0 • Drag maxima lower and wider than in 1 0 0.8 2D: → continuous spectrum, even for D/D 0.6 trapped lee waves 2al = 4.0 0.4 • Agreement with numerical simulations 0.2 requires considering both D1 and D2 0.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 π • D /D substantially higher than in 2D → l1H/ 2 1 non-hydrostatic effects more important Workshop on drag processes – 12-15 8 Sb 2016 Department of Meteorology Case 1 (2D): Flow field w/(Uh /a) for ll = 2.0/ 0 12 la1 = 2

lH1 /π = 0.5 DD 12 = .0/ 08 Propagating waves dominate

lH1 /π = 0.7 DD 21 = .0/ 06 Trapped lee waves dominate

Workshop on drag processes – 12-15 9 Sb 2016 Department of Meteorology Case 1 (3D): Resonant trapped lee wave field

w/(Uh0/a) at z=H/2 1al = 2 for ll 12 = 2.0/ lH1 /π = 0.5

30

20 1al = 5

10 30

0 20 y/H

-10 10

-20 0 y/H

-30 -10

0 10 20 30 40 50 60 70 -20 x/H

-30

0 10 20 30 40 50 60 70 “Ship-wave” pattern x/H

Workshop on drag processes – 12-15 10 Sb 2016 Department of Meteorology Case 2 (2D): Drag

Results for 2 Hl = 5.0

5 Numerical simulations hl 02 = .0 01 (b)

4 5 D /D 1 0 (c) D2/D0 3 D/D 4 la= 2 0

2 0 FLEX D1/D0

D

/ D /D = 3 2 0

D 2 la 1 2 D/D0 0

FLEX D /

1 D 2

0 1 10-1 100 101 Fr 0 -1 0 1 10 10 10 5 (d) Fr 4 D /D 1 0 • D /D may be large (~3) D2/D0 2 0 = 3 la2 0.5 D/D0

0 FLEX Fr ≈ 1 D • Single drag maximum exists at /

D 2 • Agreement with numerical simulations 1 requires considering both D1 and D2

0 -1 0 1 • D2/D1 increases as l2a decreases 10 10 10 Fr

Workshop on drag processes – 12-15 11 Sb 2016 Department of Meteorology Case 2 (3D): Drag

3.0 3.0 Internal (theory) 2.5 Lee wave (theory) 2.5 Total drag (theory) Internal (theory) Numerical Lee wave (theory) = 2.0 2.0 Total drag (theory) la2 2 la= 1 Numerical

0 2 1.5 0 1.5

D/D D/D

1.0 1.0

0.5 0.5

0.0 -1 0 1 0.0 10 10 10 10-1 100 101 Fr Fr

1.0 • D2/D0 may be large (~1.5) → some directional wave dispersion 0.8 Internal (theory) Lee wave (theory) Total drag (theory) • Drag maximum lower and wider than in 2D Numerical la2 = 0.5 0.6 0 → continuous spectrum of trapped lee waves

D/D 0.4 • Agreement with numerical simulations requires considering both D1 and D2 0.2 • D2/D1 substantially larger than in 2D and 0.0 -1 0 1 occur for lower l a → more non-hydrosatatic 10 10 10 2 Fr flow.

Workshop on drag processes – 12-15 12 Sb 2016 Department of Meteorology Case 2 (3D): Resonant trapped lee wave field

w/(Uh0/a) at z=H

for Fr = .0 85 2 Hl π = 5.0/ 2al =1

10

2al = 2

5 10

0 y/a 5

-5 0 y/a

-10 -5 0 5 10 15 20 x/a -10

0 5 10 15 20 “Ship wave” pattern x/a

Workshop on drag processes – 12-15 13 Sb 2016 Department of Meteorology Drag coefficient

h π = 0 D D h 2 c = = lh 2D obstacle + ax )/(1 D 2 0 )2/1( ρ0 AU length D0 2

h h = 0 D D π 3D obstacle 2 2 2/3 c = = lh [ ++ ayax )/()/(1 ] D 2 0 )2/1( ρ0 AU D0 4

Since for realistic atmospheric and orographic parameters, lh0=0.1~0.5, multiplying factor relating D/D0 and cD is typically 0.1~0.8

cD may easily be of O(1), especially for 2D mountains. This is comparable to turbulent form drag on obstacles in non- stratified flow.

Workshop on drag processes – 12-15 14 Sb 2016 Department of Meteorology More details Teixeira, Argain and Miranda (2013a), QJRMS, 139, 964-981 Teixeira, Argain and Miranda (2013b), JAS, 70, 2930-2947

Acknowledgements

• European Commission, through Marie Curie Career Integration Grant GLIMFLO, contract PCIG13-GA-2013-618016

Special Issue of Frontiers in Earth Science “The Atmosphere over Mountainous Regions” http://journal.frontiersin.org/researchtopic/3327/the-atmosphere- over-mountainous-regions

Workshop on drag processes – 12-15 15 Sb 2016 Department of Meteorology Summary

• 2D waves trapped in a layer may have multiple modes, 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, higher than uniform-flow hydrostatic reference value

• D2/D1 increases as l2a decreases and as mountain becomes more 3D – non-hydrostatic effects. Trapped lee wave drag maximized for l2a = O(1): wavelength of trapped lee waves matches mountain width • 3D trapped lee waves produce less drag, and drag maxima are lower and wider: continuous wave spectrum – “ship wave” pattern.

• Trapped lee waves give substantial contribution to low-level drag, may be counted mistakenly as blocking drag or turbulent form drag (different dependence)

Workshop on drag processes – 12-15 16 Sb 2016