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 g dθ N 2 = Energy Flux θ0 dz Scorer parameter 2/1 N 2 1 d2 U = − 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 DU0 = ρ0 lh0 D= ρ U lah 1+ (x / a ) [1+ (x / a ) + ( y / a ) ] 4 0 4 0 0 Workshop on drag processes – 12-15 3 Sb 2016 Department of Meteorology Linear theory +∞+∞ •Linearization, Boussinesq approximation + w(,,) x y z= wˆ(,,) k k z ei() k1 x k 2 y dk dk •Inviscid, nonrotating, stationary, uniform flow ∫ ∫ 1 2 1 2 −∞−∞ d2 wˆ k2 + k 2 + 1 2 2 −2 ˆ = 2 2 ()l k1 w 0 Taylor-Goldstein equation dz k1 ˆ Boundary conditions: wˆ( z = 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 g′ = g 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 D= p( z = 0) dxdy = 8π 2 Im k pˆ( z= 0) hˆ* dk dk ∫ ∫ ∫ ∫ 1 1 2 solutions for wˆ −∞−∞ ∂x −∞ 0 Case 1 Propagating wave drag (2D) l2 k|| hˆ 22 mm DU= 4πρ 2 12 dk |k|<l 10∫ 2 2 22 2 ()n()si0 m1 mH 121+ m os mH c Trapped lee wave drag (2D) m2 ()() k n k = π2 ρ 2 ˆ 2 1 j2 j D2 4 0U∑| h( k j ) | l2<|k|<l1 j 1+ n2 ( kj ) H m1 () k j Resonance condition (2D) tan[]m1 ( kj ) H = − n2 () k j π π Depends on = ρ 2 2 2 2 Drag normalized by D 0 0 U l 1 h 0 or D0 = ρ0 U l1 ah 0 4 4 l1 H l2/ l 1 l1 a Workshop on drag processes – 12-15 5 Sb 2016 Department of Meteorology Case 2 Propagating wave drag (2D) l2 2ˆ 2 2 k| h| ( m2 H)( kH ) DU1 = 4πρ0 dk |k|<l2 ∫ −2 2 2 2 0 [kHcosh( kH ) − Fr sinh(kH )] + ( m2 H ) sinh (kH ) Trapped lee wave drag (2D) 2 2 ˆ −2 2 2 kLL h() k {[Fr − n2 ()() kL H] − kL H } = π2 ρ 2 D2 4 0U 2 −1 −2 − 2 |k|>l2 (kL H )[][ H+ n2 ( kL ) + H 1 + n2 ( kL ) H − Fr][ Fr − n2 () kL H ] 6 k H = L Resonance condition (2D) tanh()kL H −2 Fr − n2 () kL H Depends on π π = ρ 2 2 2 2 Drag normalized by D 0 0 U l 2 h 0 or D0 = ρ0 U l2 ah 0 U l H l a 4 4 Fr = 2 2 g′ H 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 l/ l = 0.2 0 2 1 la1 = 2 lH1 /π = 0.5 DD2/ 1 = 0.08 Propagating waves dominate lH1 /π = 0.7 DD1/ 2 = 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 l1 a = 2 for l2/ l 1 = 0.2 lH1 /π = 0.5 30 20 l1 a = 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 D1/D0 • 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 l2 H /π = 0.5 l2 a =1 10 l2 a = 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 1+ (x / a ) D 2 0 )2/1( ρ0UAlength D0 2 h h = 0 D D π 3D obstacle 2 2 2/3 c = = lh [1+ (x / a ) + ( y / a ) ] D 2 0 )2/1( ρ0UA 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.