✶ ■♥tr♦❞✉❝✐♥❣ t❤❡ ❵❆❇❈ ♠♦❞❡❧✬ ❢♦r t❤❡ ❡✣❝✐❡♥t st✉❞② ♦❢ ❝♦♥✈❡❝t✐✈❡✲s❝❛❧❡ ❞❛t❛ ❛ss✐♠✐❧❛t✐♦♥ t❡❝❤♥✐q✉❡s ❘♦ss ❇❛♥♥✐st❡r✱ r✳♥✳❜❛♥♥✐st❡r❅r❡❛❞✐♥❣✳❛❝✳✉❦ ◆❈❊❖✱ ❉❡♣t✳ ♦❢ ▼❡t❡♦r♦❧♦❣②✱ ❯♥✐✈✳ ♦❢ ❘❡❛❞✐♥❣✱ ❯❑ ▼❖❉❘❙❲ ❉❆ ❲♦r❦s❤♦♣ ✲ ✶✻✴✶✼ ▼❛② ✷✵✶✾✱ ❯♥✐✈✳ ♦❢ ▲❡❡❞s
✷ ❈♦♥✈❡❝t✐✈❡✲❙❝❛❧❡ ◆❲P ✭❦♠ ❣r✐❞ s❝❛❧❡✮ ❙♦♠❡ ❝❤❛❧❧❡♥❣❡s ❝♦♠♣❛r❡❞ t♦ ❧❛r❣❡✲s❝❛❧❡ ◆❲P • ❉②♥❛♠✐❝s✿ ✕ ❍✐❣❤❧② ♥♦♥✲❧✐♥❡❛r✱ ✸❉✱ ❝♦♠♣r❡ss✐❜❧❡ ✢♦✇ ✇✐t❤ s❤♦rt ♣r❡❞✐❝t❛❜✐❧✐t② t✐♠❡s❝❛❧❡✳ ✕ ❆❝♦✉st✐❝ ♠♦❞❡s ❛♣♣❡❛r ✐♥ t❤❡ s♦❧✉t✐♦♥ t♦ ♠♦❞❡❧ ❡q✉❛t✐♦♥s ✭❝❧❡✈❡r s♦❧✈❡rs ♥❡❡❞❡❞✮✳ ✕ ◆❡❛r✲❤②❞r♦st❛t✐❝ ❜❛❧❛♥❝❡ ✭♥♦♥✲❝♦♥✈❡❝t✐♥❣ r❡❣✐♦♥s✮✳ ✕ ◆♦♥✲❤②❞r♦st❛t✐❝ ✢♦✇ ✭❝♦♥✈❡❝t✐✈❡ r❡❣✐♦♥s✮✳ ✕ ❙✐❣♥✐✜❝❛♥t ❛❣❡♦str♦♣❤✐❝ ❝♦♠♣♦♥❡♥t ♦♥ t❤❡ s♠❛❧❧ s❝❛❧❡✳ ✕ ❙♠❛❧❧✲s❝❛❧❡ q✉❛♥t✐t✐❡s ❜❡❝♦♠❡ ✐♠♣♦rt❛♥t ✭❡✳❣✳ ✈❡rt✐❝❛❧ ♠♦t✐♦♥✱ ❤②❞r♦♠❡✲ t❡♦rs✮✳ • ■♠♣♦rt❛♥t ♦❜s❡r✈❛t✐♦♥s✿ ✕ ❘❛❞❛r r❡✢❡❝t✐✈✐t②✱ ■❘✴▼❲ s♦✉♥❞❡rs✴✐♠❛❣❡rs✳ ✕ ❍✐❣❤✲❞❡♥s✐t② ♦❜s❡r✈❛t✐♦♥s ✭❝♦rr❡❧❛t❡❞ ❡rr♦rs✮✳
✸ ❚❤❡ ❵❆❇❈✬ ❙✐♠♣❧✐✜❡❞ ▼♦❞❡❧ ❘❡s❡❛r❝❤ ❜r✐❡❢ ❢♦r s✐♠♣❧✐✜❡❞ ♠♦❞❡❧ t♦ ❢❛❝✐❧✐t❛t❡ ❝♦♥✈❡❝t✐✈❡✲s❝❛❧❡ ❉❆ r❡s❡❛r❝❤ • ❙t❛rt✐♥❣ ♣♦✐♥t✿ t❤❡ ✸❉ ❝♦♠♣r❡ss✐❜❧❡ ❊✉❧❡r ❡q✉❛t✐♦♥s✳ • ◆♦♥✲❤②❞r♦st❛t✐❝✱ ❝♦♠♣r❡ss✐❜❧❡✳ • ❯♥❜❛❧❛♥❝❡❞ ♦♥ s♠❛❧❧ s❝❛❧❡s✱ ❜❛❧❛♥❝❡❞ ♦♥ ❧❛r❣❡ s❝❛❧❡s✳ • ❈♦♥s❡r✈❡ ♠❛ss ❛♥❞ ❡♥❡r❣②✿ d � d � ρ ( x, z, t ) dxdz = 0 E ( x, z, t ) dxdz = 0 . dt dt • ❈❤❡❛♣ t♦ r✉♥✿ ✕ ■♥t❡❣r❛t❡❞ ✇✐t❤ ❛♥ ❡①♣❧✐❝✐t ♥✉♠❡r✐❝❛❧ s❝❤❡♠❡✳ ✕ ✷❉ ✭❧♦♥❣✐t✉❞❡✴❤❡✐❣❤t ❞♦♠❛✐♥✮✳ • ❉r② ❞②♥❛♠✐❝s ✭❝✉rr❡♥t❧②✮✳
✹ ❆❇❈ ▼♦❞❡❧✿ ❚✉♥❛❜❧❡ P❛r❛♠❡t❡rs✱ ❆✱ ❇✱ ❈ • u = ( u, v, w ) ✳ ρ ′ ∂u ∂t + B u · ∇ u + C∂ ˜ ∂x − fv = 0 ρ ′ s❝❛❧❡❞ ❞❡♥s✐t② ♣❡rt✿ ˜ ρ ′ = ( ρ − ρ 0 ) /ρ 0 ✳ • ˜ ∂v ∂t + B u · ∇ v + fu = 0 • b ′ ❜✉♦②❛♥❝②✿ b ′ = ( θ − [ θ R + θ 0 ( z )]) g/θ R ✳ ρ ′ ∂w ∂t + B u · ∇ w + C∂ ˜ ∂z − b ′ = 0 • A ♣✉r❡ ❣r❛✈✐t② ✇❛✈❡ ❢r❡q✉❡♥❝②✳ ρ ′ ∂ ˜ ∂t + B ∇ · (˜ ρ u ) = 0 • B ♠♦❞✉❧❛t✐♦♥ 0 < B ≤ 1 ✱ t♦ r❡❞✉❝❡ ∂b ′ ∂t + B u · ∇ b ′ + A 2 w = 0 ❛❝♦✉st✐❝ ✇❛✈❡ ❢r❡q✉❡♥❝✐❡s✳ √ p ′ = Cρ 0 ˜ ρ ′ BC ♣✉r❡ ❛❝♦✉st✐❝ ✇❛✈❡ s♣❡❡❞✳ • ▼♦st ❡①♣❡r✐♠❡♥ts✿ A = 0 . 02 s − 1 , B = 0 . 01 , C = 10000 m 2 s − 2 . ❙❡❡ P❡tr✐❡✱ ❇❛♥♥✐st❡r✱ ❛♥❞ ❈✉❧❧❡♥ ✭✷✵✶✼✮✳
✺ ▼♦❞❡❧ ❜❡❤❛✈✐♦✉r ✖ ❧✐♥❡❛r ♠♦❞❡s ❍♦r✐③♦♥t❛❧ ❣r❛✈✐t② ❣r♦✉♣ ✇❛✈❡ s♣❡❡❞ ❍♦r✐③♦♥t❛❧ ❛❝♦✉st✐❝ ❣r♦✉♣ ✇❛✈❡ s♣❡❡❞ (a) (b) Gravity wave speeds (horizontal) Acoustic wave speeds (horizontal) 160 1000 BC = 10 4 A = 0 . 002 BC = 10 5 A = 0 . 02 140 BC = 10 6 A = 0 . 2 800 120 100 600 speed (m/s) speed (m/s) 80 400 60 40 200 20 0 0 0 50 100 150 200 250 300 350 0 50 100 150 200 250 300 350 horizontal wavenumber index horizontal wavenumber index √ ❘❡❝❛❧❧✿ BC ♣✉r❡ ❛❝♦✉st✐❝ ✇❛✈❡ s♣❡❡❞
✻ ▼♦❞❡❧ ❜❡❤❛✈✐♦✉r ✖ ✐❞❡❛❧✐s❡❞ ❧❛r❣❡✲s❝❛❧❡ ρ ′ ✮ ❛♥❞ ❤♦r✐③✳ ✇✐♥❞s ( u, v ) ❙❝❛❧❡❞ ❞❡♥s✐t② ♣❡rt✳ ✭ ˜ (a) t=0s (b) t=10800s (three hours) (c) t=21600s (six hours) ρ pert. and horiz. winds for t = 0h ρ pert. and horiz. winds for t = 3h ρ pert. and horiz. winds for t = 6h ˜ ˜ ˜ 14 14 14 12 12 12 Vertical distance (km) Vertical distance (km) Vertical distance (km) 10 10 10 8 8 8 6 6 6 4 4 4 2 2 2 100 200 300 400 500 100 200 300 400 500 100 200 300 400 500 Longitudinal distance (km) Longitudinal distance (km) Longitudinal distance (km) −0.0008 −0.0004 0.0000 0.0004 0.0008 0.0012 0.0016 0.0020 0.0024
✼ ▼♦❞❡❧ ❜❡❤❛✈✐♦✉r ✖ ❡①❛♠♣❧❡ ✐♥t❡❣r❛t✐♦♥ w ✭ t = 0 ✮ w ✭ t = 3 ❤ ✮ w ✭ t = 6 ❤ ✮ w for t = 0h w for t = 3h w for t = 6h 14 14 14 12 12 12 Vertical distance (km) Vertical distance (km) Vertical distance (km) 10 10 10 8 8 8 6 6 6 4 4 4 2 2 2 100 200 300 400 500 100 200 300 400 500 100 200 300 400 500 Longitudinal distance (km) Longitudinal distance (km) Longitudinal distance (km) ∂ b ′ / ∂ z + A 2 for t = 3h min=-0.00, max=0.00, mean=0.00040, rms=0.00015 0.00150 14 0.00125 12 0.00100 Vertical distance (km) 10 0.00075 ∂ ∂z [ b 0 ( z ) + b ′ ( x, z, t )] = 8 0.00050 ❊✛❡❝t✐✈❡ ❜✉♦②❛♥❝② A 2 + ∂ 6 0.00025 ∂z b ′ ( x, z, t ) 0.00000 4 −0.00025 2 −0.00050 100 200 300 400 500 Longitudinal distance (km) tracer for t = 3h min=0.00, max=0.42, mean=0.00472, rms=0.02510 0.48 14 ❚r❛❝❡r 0.42 12 0.36 Vertical distance (km) 10 0.30 8 0.24 6 0.18 4 0.12 2 0.06 0.00 100 200 300 400 500 Longitudinal distance (km)
✽ ▼♦❞❡❧ ❜❡❤❛✈✐♦✉r ✖ ✐♠❜❛❧❛♥❝❡ ●❡♦str♦♣❤✐❝ ❛♥❞ ❤②❞r♦st❛t✐❝ ✐♠❜❛❧❛♥❝❡s Geostrophic and hydrostatic imbalance 0.9 0.12 � � ρ ′ C ∂ ˜ RMS ∂x − fv 0.8 ❣❡♦✳ ✐♠❜❛❧✳ = 0.10 � � � � ρ ′ C ∂ ˜ 0.7 RMS + RMS fv Geostrophic imbalance Hydrostatic imbalance ∂x 0.6 0.08 0.5 0.06 0.4 >100.0km >100.0km � ∂z − b ′ � ρ ′ C ∂ ˜ RMS >10.0km >10.0km 0.3 0.04 >1.0km >1.0km ❤②❞r♦✳ ✐♠❜❛❧✳ = � � � � 0.2 ρ ′ C ∂ ˜ RMS + RMS b ′ 0.02 ∂z 0.1 0.0 0.00 0.0 0.5 1.0 1.5 2.0 2.5 3.0 time (hours)
✾ ■♥tr♦❞✉❝✐♥❣ t❤❡ ❞❛t❛ ❛ss✐♠✐❧❛t✐♦♥ ❝❛♣❛❜✐❧✐t② ❉❡s✐❣♥ ♦❢ t❤❡ ❉❆ • ❚②♣❡ ♦❢ ❉❆✿ ✈❛r✐❛t✐♦♥❛❧ ✭❝✉rr❡♥t❧② ✸❉❱❛r ❛♥❞ ✸❉❋●❆❚ ✐♠♣❧❡♠❡♥t❡❞✮ J ( δ x ) = 1 2 δ x T B − 1 δ x + 1 � T R − 1 ( • ) y t − H t [ M t ( x b )] − H t M t δ x � 2 δ x = x − x b . • ❇❛❝❦❣r♦✉♥❞ ❡rr♦r ❝♦✈❛r✐❛♥❝❡s ✭ B ✮✿ ✕ ◆❡❡❞ t♦ ❜❡ ♠♦❞❡❧❧❡❞ ❡✣❝✐❡♥t❧② ✭❝♦♥tr♦❧ ✈❛r✐❛❜❧❡ tr❛♥s❢♦r♠✱ δ x = U δ χ ✱ � δ x δ x T � � δ χ δ χ T � = B ✱ = I ✮✱ ✕ ▲❛r❣❡ r❛♥❣❡ ♦❢ ♦♣t✐♦♥s ✭❡✳❣✳ ❡①♣❧♦✐t✐♥❣ ❣❡♦♣❤②s✐❝❛❧ ❜❛❧❛♥❝❡s✱ ♦♣t✐♦♥s ❢♦r ❤♦r✐③♦♥t❛❧ ❛♥❞ ✈❡rt✐❝❛❧ tr❛♥s❢♦r♠s✮✳ • ❖❜s❡r✈❛t✐♦♥s✿ ✕ ❋❧❡①✐❜❧❡ ♦❜s❡r✈❛t✐♦♥ ♥❡t✇♦r❦ ✭❡✳❣✳ ❛♥② q✉❛♥t✐t②✱ ❛♥② t✐♠❡✴♣♦s✐t✐♦♥✱ ❛❧❧♦✇ ♦✉t❧✐❡r ♦❜s❡r✈❛t✐♦♥s✮✳
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