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

Transcription

1 Trapped lee waves: a currently neglected source of low-level orographic drag Miguel A. C. Teixeira, Jose L. Argain, Pedro M. A. Miranda 3, University of Reading, Reading, UK Physics, University of Algarve, Faro, Portugal 3 Instituto om Luiz (IL), University of Lisbon, Lisbon, Portugal

2 Trapped lee waves N g dθ θ dz Non-hydrostatic Energy Flux Scorer parameter / N d U l U dz U Workshop on drag processes -

3 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? h Bell-shaped and 3 circular mountains h h h + ( x / a) + ( x / a) + ( y / a ρu lh [ ) ] 3/ Linear, hydrostatic, non-rotating, constant l limit π π ρu lah Workshop on drag processes - 3

4 Linear theory Linearization, Boussinesq approximation Inviscid, nonrotating, stationary, uniform flow ˆ d w k + k + ( l k ) ˆ w dz k + + i( kx+ k y) wˆ ( k, k, z) e dk w( x, y, z) dk Taylor-Goldstein equation ŵ pˆ Boundary conditions: continuous at zh w( z ) iuk hˆ ˆ Waves propagate energy upward or decay as z -layer atmospheres Scorer (99) Vosper () l l N U N U Case Case θ g g θ θ Workshop on drag processes -

5 + + Gravity wave drag + + h * p( z ) dxdy 8π Im k pˆ( z ) h ˆ dkdk x pˆ determined from solutions for ŵ Case Propagating wave drag () ˆ U dk m cos ( mh) m sin ( mh) l k h mm πρ + k <l Trapped lee wave drag () m ( k j ) n ( k j ) π ρu hˆ( k j ) j + n ( k ) H j l < k <l Resonance condition () tan [ m ( k ) H ] j m ( k n ( k j j ) ) rag normalized by π ρu l h or π ρu l ah epends on H l / l l a l Workshop on drag processes -

6 Case Propagating wave drag () πρ U l [ kh cosh( kh ) Fr sinh( kh )] k hˆ ( m H )( kh ) + ( m H ) sinh ( kh ) dk k <l Trapped lee wave drag () kl hˆ( kl ) {[ Fr n ( kl ) H ] ( klh ) } [ H + n ( k )] + H [ + n ( k ) H Fr ][ Fr n ( k ) H ] π ρu ( k H ) L L L L k >l Resonance condition () tanh ( k H ) L klh Fr n ( k L ) H 6 rag normalized by π ρu l h or π ρu l ah Fr epends on U g H l H l a Workshop on drag processes - 6

7 (a) la la / 6 3 la l a. FLEX / / / (c) /..... Results for l H/π Case (): rag FLEX / / / l l H/π / l. la la Numerical simulations l h. (b) / 3 FLEX / / / l H/π / may be large (~3) rag maxima coincide with establishment of trapped lee wave modes Agreement with numerical simulations requires considering both and / increases as l a decreases Workshop on drag processes - 7

8 Case (3): rag la la / Numerical Total drag (theory) Internal (theory) Lee wave (theory) la la / Numerical Total drag (theory) Internal (theory) Lee wave (theory) l H/π l H/π la l a. / l H/π Numerical Total drag (theory) Internal (theory) Lee wave (theory) / may be large (~) some directional wave dispersion rag maxima lower and wider than in : continuous spectrum, even for trapped lee waves Agreement with numerical simulations requires considering both and / substantially higher than in non-hydrostatic effects more important Workshop on drag processes - 8

9 Case (): Flow field w/(uh /a) for l / l. la lh / π. / Propagating waves dominate.8 lh / π.7. 6 / Trapped lee waves dominate Workshop on drag processes - 9

10 Case (3): Resonant trapped lee wave field w/(uh /a) at zh/ for l l. lh / π. / l a 3 l a 3 y/h - y/h x/h x/h Ship-wave pattern Workshop on drag processes -

11 la / 3 Results for (b) Case (): rag l H. / / / FLEX Numerical simulations l h. la / 3 (c) / / / FLEX la. / - 3 (d) Fr / / / FLEX - Workshop on drag processes - Fr - Fr / may be large (~3) Single drag maximum exists at Fr Agreement with numerical simulations requires considering both and / increases as l a decreases

12 Case (3): rag la /... Internal (theory) Lee wave (theory) Total drag (theory) Numerical la /... Internal (theory) Lee wave (theory) Total drag (theory) Numerical Fr. - Fr la. / Fr Internal (theory) Lee wave (theory) Total drag (theory) Numerical / may be large (~.) some directional wave dispersion rag maximum lower and wider than in continuous spectrum of trapped lee waves Agreement with numerical simulations requires considering both and / substantially larger than in and occur for lower l a more non-hydrosatatic flow. Workshop on drag processes -

13 Case (3): Resonant trapped lee wave field w/(uh /a) at zh for Fr. 8 l H / π. l a l a y/a y/a x/a x/a Ship wave pattern Workshop on drag processes - 3

14 rag coefficient obstacle h h + ( x / a) c (/ ) ρ U Alength π lh 3 obstacle h h [ + ( x / a) + ( y / a) ] 3/ c (/ ) ρ U A π lh Since for realistic atmospheric and orographic parameters, lh.~., multiplying factor relating / and c is typically.~.8 c may easily be of O(), especially for mountains. This is comparable to turbulent form drag on obstacles in nonstratified flow. Workshop on drag processes -

15 More details Teixeira, Argain and Miranda (3a), QJRMS, 39, Teixeira, Argain and Miranda (3b), JAS, 7, Acknowledgements European Commission, through Marie Curie Career Integration Grant GLIMFLO, contract PCIG3-GA Special Issue of Frontiers in Earth Science The Atmosphere over Mountainous Regions Workshop on drag processes -

16 Summary waves trapped in a layer may have multiple modes, waves trapped at temperature inversion may only have single mode ue 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 / increases as l a decreases and as mountain becomes more 3 non-hydrostatic effects. Trapped lee wave drag maximized for l a O(): wavelength of trapped lee waves matches mountain width 3 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 - 6