1/ 12 11 – Waves and Instabilities Waves Rossby waves Waves Rossby waves 2/ 12 I potential vorticity equation for layered model Dq Dt =0 , q = ζ - f 0 H h + f 0 + β y with relative vorticity ζ =(g /f 0 )∇ 2 h I linearized version (D /Dt → ∂/∂ t ) D Dt g f 0 ∇ 2 h - f 0 H h + f 0 + β y = D Dt g f 0 ∇ 2 h - f 0 H h + β v =0 ∂ ∂ t g f 0 ∇ 2 h - f 0 H h + β g f 0 ∂ h ∂ x ≈ 0 → ∂ ∂ t ( ∇ 2 h - R -2 h ) + β ∂ h ∂ x =0 with Rossby radius R = √ gH /|f | I ”vorticity wave” h = A exp i (k 1 x + k 2 y - ωt )= A exp i (k · x - ωt ) (-i ω) ( (i k ) 2 A exp i (..) - R -2 A exp i (..) ) + β (ik 1 )A exp i (..)=0 - ω ( -(k ) 2 - R -2 ) + β k 1 =0 ω = - β k 1 k 2 + R -2 with k 2 = k 2 1 + k 2 2 =(k ) 2
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Waves Rossby waves 2/ 12 11 { Waves and Instabilities · Waves Rossby waves 3/ 12 I "vorticity wave" !Rossby wave dispersion relation k 1 k2 + R 2 with Rossby radius R = p gH=jf jand
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1/ 12
11 – Waves and Instabilities
WavesRossby waves
Waves Rossby waves 2/ 12
I potential vorticity equation for layered model
Dq
Dt= 0 , q = ζ − f0
Hh + f0 + βy
with relative vorticity ζ = (g/f0)∇2h
I linearized version (D/Dt → ∂/∂t)
D
Dt
(g
f0∇2h − f0
Hh + f0 + βy
)=
D
Dt
(g
f0∇2h − f0
Hh
)+ βv = 0
∂
∂t
(g
f0∇2h − f0
Hh
)+ β
g
f0
∂h
∂x≈ 0 → ∂
∂t
(∇2h − R−2h
)+ β
∂h
∂x= 0
with Rossby radius R =√gH/|f |
I ”vorticity wave” h = A exp i(k1x + k2y − ωt) = A exp i(k · x − ωt)
(−iω)((ik)2A exp i(..)− R−2A exp i(..)
)+ β(ik1)A exp i(..) = 0
− ω(−(k)2 − R−2
)+ βk1 = 0
ω = − βk1k2 + R−2
with k2 = k21 + k2
2 = (k)2
Waves Rossby waves 3/ 12
I ”vorticity wave” → Rossby wave dispersion relation
ω = − βk1k2 + R−2
with Rossby radius R =√gH/|f | and k2 = k2
1 + k22 = (k)2
I slow, only present with planetary vorticity gradient df /dy = β
I for k1 > 0 phase propagation speed c = ω/k is negative
for k1 < 0 phase propagation speed c = ω/k is positive
→ phase propagation is always westward
0.05 0.00 0.05k1 [1/km]
2
1
0
1
2
1/R-1/R
R=50km
ω(k1 ) [cycles/year] for k2 =0
0.05 0.00 0.05k1 [1/km]
0.05
0.00
0.05
ω(k1 ,k2 ) [cycles/year]
2.25
1.50
0.75
0.00
0.75
1.50
2.25
Waves Rossby waves 4/ 12
I ”vorticity wave” → Rossby wave dispersion relation
ω = − βk1k2 + R−2
with Rossby radius R =√gH/|f | and k2 = k2
1 + k22 = (k)2
I meridional group velocity of Rossby waves
cyg =∂ω
∂k2= −(−1)
βk1(k2 + R−2)2
2k2 =2βk1k2
(k2 + R−2)2
I zonal group velocity of Rossby waves
cxg =∂ω
∂k1= −(−1)
βk1(k2 + R−2)2
2k1 −β
k2 + R−2
=2βk2
1
(k2 + R−2)2− β(k2 + R−2)
(k2 + R−2)2=
2βk21 − βk2
1 − βk22 − βR−2
(k2 + R−2)2
= βk21 − k2
2 − R−2
(k2 + R−2)2
Waves Rossby waves 5/ 12
I Group velocity of Rossby waves
cxg = βk21 − k2
2 − R−2
(k2 + R−2)2, cyg =
2βk1k2
(k2 + R−2)2
0.05 0.00 0.05k1 [1/km]
0.05
0.00
0.05
k2 [1
/km
]
R=50kmc xg (k1 ,k2 ) [cm/s]
2.25
1.50
0.75
0.00
0.75
1.50
2.25
0.05 0.00 0.05k1 [1/km]
0.05
0.00
0.05
c yg (k1 ,k2 ) [cm/s]
2.25
1.50
0.75
0.00
0.75
1.50
2.25
I direction of cxg westward for small k1, eastward for large k1
I for k2 = 0 cxg = 0 at k1 = ±1/R → ωmax = −βR/2
I direction of cyg always opposite to k2
Waves Rossby waves 6/ 12
I Rossby wave dispersion relation
ω = − βk1k2 + R−2
, cxg = βk21 − k2
2 − R−2
(k2 + R−2)2, cyg =
2βk1k2
(k2 + R−2)2
with Rossby radius R =√gH/|f | and k2 = k2
1 + k22 = (k)2
I long wave limit for λ� R or k � 1/R or kR → 0
ωkR→0
= −βk1R2 , cxgkR→0
= −βR2 , cygkR→0
= 0
→ westward phase and energy propagation
→ no dispersion: ckR→0
= cxg (for a wave with k2 = 0)
I start again with PV equation and neglect relative vorticity ζ = ∇2h