tr t r t - - PowerPoint PPT Presentation
tr t r t t st ts t sst tqs
✶
❘♦ss ❇❛♥♥✐st❡r✱ r✳♥✳❜❛♥♥✐st❡r❅r❡❛❞✐♥❣✳❛❝✳✉❦ ◆❈❊❖✱ ❉❡♣t✳ ♦❢ ▼❡t❡♦r♦❧♦❣②✱ ❯♥✐✈✳ ♦❢ ❘❡❛❞✐♥❣✱ ❯❑ ▼❖❉❘❙❲ ❉❆ ❲♦r❦s❤♦♣ ✲ ✶✻✴✶✼ ▼❛② ✷✵✶✾✱ ❯♥✐✈✳ ♦❢ ▲❡❡❞s
✷
❙♦♠❡ ❝❤❛❧❧❡♥❣❡s ❝♦♠♣❛r❡❞ t♦ ❧❛r❣❡✲s❝❛❧❡ ◆❲P
✕ ❍✐❣❤❧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✮✳
✕ ❘❛❞❛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❝❤
d dt
d dt
✕ ■♥t❡❣r❛t❡❞ ✇✐t❤ ❛♥ ❡①♣❧✐❝✐t ♥✉♠❡r✐❝❛❧ s❝❤❡♠❡✳ ✕ ✷❉ ✭❧♦♥❣✐t✉❞❡✴❤❡✐❣❤t ❞♦♠❛✐♥✮✳
✹
∂u ∂t + Bu · ∇u + C∂ ˜ ρ′ ∂x − fv = 0 ∂v ∂t + Bu · ∇v + fu = 0 ∂w ∂t + Bu · ∇w + C∂ ˜ ρ′ ∂z − b′ = 0 ∂ ˜ ρ′ ∂t + B∇ · (˜ ρu) = 0 ∂b′ ∂t + Bu · ∇b′ + A2w = 0 p′ = Cρ0˜ ρ′
ρ′ s❝❛❧❡❞ ❞❡♥s✐t② ♣❡rt✿ ˜ ρ′ = (ρ − ρ0)/ρ0✳
❛❝♦✉st✐❝ ✇❛✈❡ ❢r❡q✉❡♥❝✐❡s✳
BC ♣✉r❡ ❛❝♦✉st✐❝ ✇❛✈❡ s♣❡❡❞✳ ▼♦st ❡①♣❡r✐♠❡♥ts✿ A = 0.02 s−1, B = 0.01, C = 10000 m2s−2. ❙❡❡ P❡tr✐❡✱ ❇❛♥♥✐st❡r✱ ❛♥❞ ❈✉❧❧❡♥ ✭✷✵✶✼✮✳
✺
❍♦r✐③♦♥t❛❧ ❣r❛✈✐t② ❣r♦✉♣ ✇❛✈❡ s♣❡❡❞ ❍♦r✐③♦♥t❛❧ ❛❝♦✉st✐❝ ❣r♦✉♣ ✇❛✈❡ s♣❡❡❞
50 100 150 200 250 300 350 horizontal wavenumber index 200 400 600 800 1000 speed (m/s)
Acoustic wave speeds (horizontal) BC = 104 BC = 105 BC = 106
50 100 150 200 250 300 350 horizontal wavenumber index 20 40 60 80 100 120 140 160 speed (m/s)
Gravity wave speeds (horizontal) A = 0. 002 A = 0. 02 A = 0. 2
(b) (a)
❘❡❝❛❧❧✿ √ BC ♣✉r❡ ❛❝♦✉st✐❝ ✇❛✈❡ s♣❡❡❞
✻
❙❝❛❧❡❞ ❞❡♥s✐t② ♣❡rt✳ ✭˜ ρ′✮ ❛♥❞ ❤♦r✐③✳ ✇✐♥❞s (u, v)
100 200 300 400 500 Longitudinal distance (km) 2 4 6 8 10 12 14 Vertical distance (km) ˜ ρ pert. and horiz. winds for t = 6h
−0.0008 −0.0004 0.0000 0.0004 0.0008 0.0012 0.0016 0.0020 0.0024
100 200 300 400 500 Longitudinal distance (km) 2 4 6 8 10 12 14 Vertical distance (km) ˜ ρ pert. and horiz. winds for t = 3h 100 200 300 400 500 Longitudinal distance (km) 2 4 6 8 10 12 14 Vertical distance (km) ˜ ρ pert. and horiz. winds for t = 0h
(a) t=0s (b) t=10800s (three hours) (c) t=21600s (six hours)
✼
w ✭t = 0✮ w ✭t = 3 ❤✮ w ✭t = 6 ❤✮
100 200 300 400 500 Longitudinal distance (km) 2 4 6 8 10 12 14 Vertical distance (km) w for t = 0h❊✛❡❝t✐✈❡ ❜✉♦②❛♥❝② ❚r❛❝❡r
100 200 300 400 500 Longitudinal distance (km) 2 4 6 8 10 12 14 Vertical distance (km) w for t = 3h 100 200 300 400 500 Longitudinal distance (km) 2 4 6 8 10 12 14 Vertical distance (km) ∂b ′/∂z + A 2 for t = 3h min=-0.00, max=0.00, mean=0.00040, rms=0.00015 −0.00050 −0.00025 0.00000 0.00025 0.00050 0.00075 0.00100 0.00125 0.00150 100 200 300 400 500 Longitudinal distance (km) 2 4 6 8 10 12 14 Vertical distance (km) tracer for t = 3h min=0.00, max=0.42, mean=0.00472, rms=0.02510 0.00 0.06 0.12 0.18 0.24 0.30 0.36 0.42 0.48 100 200 300 400 500 Longitudinal distance (km) 2 4 6 8 10 12 14 Vertical distance (km) w for t = 6h∂ ∂z [b0(z) + b′(x, z, t)]
= A2 + ∂
∂zb′(x, z, t)
✽
0.0 0.5 1.0 1.5 2.0 2.5 3.0
time (hours)
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
Geostrophic imbalance
>100.0km >10.0km >1.0km
0.00 0.02 0.04 0.06 0.08 0.10 0.12
Hydrostatic imbalance Geostrophic and hydrostatic imbalance
>100.0km >10.0km >1.0km
❣❡♦✳ ✐♠❜❛❧✳ = RMS
ρ′ ∂x − fv
ρ′ ∂x
RMS
ρ′ ∂z − b′
RMS
ρ′ ∂z
✾
❉❡s✐❣♥ ♦❢ t❤❡ ❉❆
J(δx) = 1 2δxTB−1δx + 1 2
T R−1 (•) δx = x − xb.
✕ ◆❡❡❞ t♦ ❜❡ ♠♦❞❡❧❧❡❞ ❡✣❝✐❡♥t❧② ✭❝♦♥tr♦❧ ✈❛r✐❛❜❧❡ tr❛♥s❢♦r♠✱ δx = Uδχ✱
= B✱
= I✮✱ ✕ ▲❛r❣❡ r❛♥❣❡ ♦❢ ♦♣t✐♦♥s ✭❡✳❣✳ ❡①♣❧♦✐t✐♥❣ ❣❡♦♣❤②s✐❝❛❧ ❜❛❧❛♥❝❡s✱ ♦♣t✐♦♥s ❢♦r ❤♦r✐③♦♥t❛❧ ❛♥❞ ✈❡rt✐❝❛❧ tr❛♥s❢♦r♠s✮✳
✕ ❋❧❡①✐❜❧❡ ♦❜s❡r✈❛t✐♦♥ ♥❡t✇♦r❦ ✭❡✳❣✳ ❛♥② q✉❛♥t✐t②✱ ❛♥② t✐♠❡✴♣♦s✐t✐♦♥✱ ❛❧❧♦✇ ♦✉t❧✐❡r ♦❜s❡r✈❛t✐♦♥s✮✳
✶✵
❉❡s✐❣♥ ♦❢ t❤❡ ❉❆ ✭❝♦♥t✳✮
✕ ❈❛❧✐❜r❛t❡ B✲♠❛tr✐① ❢r♦♠ tr❛✐♥✐♥❣ ❞❛t❛✳ ✕ ❚❡st s②st❡♠ ❝♦♠♣♦♥❡♥ts ✭❡✳❣✳ ❛❞❥♦✐♥t t❡sts✱ ✐♥✈❡rs❡ t❡sts✮✳ ✕ ❈♦♠♣✉t❡ ✐♠♣❧✐❡❞ ❝♦✈❛r✐❛♥❝❡s✿ B = UUT✳ ✕ ❈♦♠♣✉t❡ r❛✇ ❝♦✈❛r✐❛♥❝❡s ❢r♦♠ tr❛✐♥✐♥❣ ❞❛t❛
✳ ✕ P❡r❢♦r♠ ❵tr✉t❤✬ r✉♥✱ xt✳ ✕ ●❡♥❡r❛t❡ xb ❜② ♣❡rt✉r❜✐♥❣ xt✳ ✕ ●❡♥❡r❛t❡ yt ❜② ♣❡rt✉r❜✐♥❣ Ht[Mt(xt)]✳ ✕ ❆ss✐♠✐❧❛t❡✦ ✕ ▼❛♥❛❣❡ ❝②❝❧✐♥❣✳ ✕ ❊❛s② ✈✐s✉❛❧✐s❛t✐♦♥ ♦❢ ❛ss✐♠✐❧❛t✐♦♥ ❞✐❛❣♥♦st✐❝s✳ ✕ ❈♦♠♣❛r❡ ❛ss✐♠✐❧❛t✐♦♥ ❡①♣❡r✐♠❡♥ts ✇✐t❤ ❞✐✛❡r❡♥t s♣❡❝✐✜❝❛t✐♦♥s✳
✶✶
✷✺✷✵ ♦❜s❡r✈❛t✐♦♥s ♦❢ ˜ ρ′
✶✷
δx = Uδχ δu δv δw δ˜ ρ′ δb′ = ∂x 0 ∂y 0 0 0 I αLgb 0 0 I αLhbLgb 0 0 Lhb I Uψ
s
Uχ
s
Uw
s
U
u˜
ρ′ s
U
ub′
s
δχψ δχχ δχw δχ
u˜
ρ′
δχ
ub′
❊✳❣✳ δψ = Uψ
s δχψ,
δu = ∂xδψ ❣❡♦✳ ❜❛❧✳ δb˜ ρ′ = Lgbδψ, ❤②❞r♦✳ ❜❛❧✳ δb′ = Lhbδ˜ ρ′, ❣❡♦✳ ❜❛❧✳ s✇✐t❝❤ α = 0, 1 ❲♦✉❧❞ ❧✐❦❡ t♦ ❝♦♠♣✉t❡ t❤❡ ✐♠♣❧✐❡❞ ❝♦✈❛r✐❛♥❝❡s B = UUT.
✶✸
B = UUT = ∂xBχ∂T
x
∂yBψ∂T
y
α∂yBψLgbT α∂yBψLgbTLhbT Bw αLgbBψ∂T
y
B˜
ρ′
B˜
ρ′LhbT
αLhbLgbBψ∂T
y
LhbB˜
ρ′
LhbB˜
ρ′LhbT + B
ub′
↑ u ↑ v ↑ w ↑ ˜ ρ′ ↑ b′ ← u ← v ← w ← ˜ ρ′ ← b′ ✇❤❡r❡ B˜
ρ′ = α2LgbBψLgbT + B
u˜
ρ′, ❛♥❞ Bψ = Uψ s Uψ s T, ❡t❝✳
✶✹
❈♦✈✳ st❛ts
■♠♣❧✐❡❞ ❈♦✈✳✱ UUT ✭●❇✰✮ ■♠♣❧✐❡❞ ❈♦✈✳✱ UUT ✭●❇✲✮ ˜ ρ′ − ˜ ρ′ v′ − ˜ ρ′
✶✺
ρ′✮✿ ✇✐t❤ ❣❡♦✳ ❜❛❧✳
α = 1 ✭●❇✰✮✱ ❜❛❝❦❣r♦✉♥❞ ✇✐♥❞ ❡rr♦rs α = 1 ✭●❇✰✮✱ ❛♥❛❧②s✐s ✇✐♥❞ ❡rr♦rs
✶✻
ρ′✮✿ ✇✐t❤♦✉t ❣❡♦✳
α = 0 ✭●❇✲✮✱ ❜❛❝❦❣r♦✉♥❞ ✇✐♥❞ ❡rr♦rs α = 0 ✭●❇✲✮✱ ❛♥❛❧②s✐s ✇✐♥❞ ❡rr♦rs
✶✼
ρ′✮✿ ✇❤✐❝❤ ✐s ❜❡tt❡r❄
diff = RMSE(●❇✲) − RMSE(●❇✰) ◆❡❣❛t✐✈❡ ✈❛❧✉❡s ✐♥❞✐❝❛t❡ t❤❛t r❡♠♦✈✐♥❣ ❣❡♦✳ ❜❛❧✳ ♦♣❡r❛t♦rs ❢r♦♠ B ✭✐✳❡✳ α = 0✮ ✐♠♣r♦✈❡s t❤❡ ❛s✲ s✐♠✐❧❛t✐♦♥✳ ❜❛❝❦❣r♦✉♥❞ ✇✐♥❞ ❘▼❙❊ ❞✐✛❡r❡♥❝❡s ❜❛❝❦❣r♦✉♥❞ s❝❛❧❡❞ ❞❡♥s✐t② ❘▼❙❊ ❞✐✛❡r❡♥❝❡s
✶✽
❝♦♥✈❡❝t✐✈❡✲s❝❛❧❡ ♠♦❞❡❧✳
✕ ❖❜s❡r✈❛t✐♦♥s ♦❢ ˜ ρ′✳ ✕ ❙✇✐t❝❤ ❖◆✴❖❋❋ ❣❡♦str♦♣❤✐❝ ❝♦♥str❛✐♥t ✐♥ B✲♠❛tr✐①✳ ✕ ❙❛♠❡ ✐♥✐t✐❛❧ ❜❛❝❦❣r♦✉♥❞ st❛t❡✳ ✕ ❈②❝❧❡ ❧❡♥❣t❤ ✶ ❤♦✉r✱ ✸✵ ❝②❝❧❡s✳ ✕ ❚✇✐♥ ❡①♣❡r✐♠❡♥ts✿ ❦♥♦✇ tr✉t❤✳ ✕ ▼✐①❡❞ ✐♠♣❛❝t ♦❢ s✇✐t❝❤✐♥❣ ♦✛ ❣❡♦str♦♣❤✐❝ ❜❛❧❛♥❝❡ ✕ ♦❢t❡♥ r❡❞✉❝❡s ❘▼❙ ❡rr♦rs✳
✕ P✉❜❧✐s❤ s②st❡♠ ❛♥❞ t❡sts✳ ✕ ■♥✈❡st✐❣❛t❡ ♦t❤❡r ❜❛❧❛♥❝❡s✳ ✕ ❆❞❞✿ ✹❉✲❱❛r✱ ♠♦✐st✉r❡✳ ✕ ◆♦r♠❛❧ ♠♦❞❡✲❜❛s❡❞ B✲♠❛tr✐①✳ ✕ ❘❡❞✉❝❡❞✲r❛♥❦ ❑❛❧♠❛♥ ❋✐❧t❡r ✳ ✳ ✳