Examining Vortex Rossby Wave (VRW) dispersion relations

with numerical experiments

Ting-Chi Wu

MPO673 Vortex Dynamics Project final report

2011/04/28

Motivation and Background• Montgomery and Kallenbach (1997) developed a theory for vortex

rossby wave (VRW) with dispersion relationship and showed the outward propagating banded features.

• McWilliams et al (2003) generalized MK97’s work by extending in finite Rossby-number regimes.

• Cobosiero et al (2009) compared theoretical VRW dispersion relation with observation data (Radar) and have good agreement.

• However, previous studies only “qualitatively” address the propagation feature of VRW in theory, however, “quantitative” verification against the theory is needed. Comparing with observation requires a few assumptions while in numerical experiment parameter can be assigned to your like.

• Therefore, numerical model based on linearized shallow water equation in Nolan et al (2001) is used to generate VRW and compare the model results with theory provided by MK97.

Linearized Asymmetric Shallow-Water Model

€

DuDt

−v 2

r− fv + g

∂h∂r

= 0

DvDt

+uvr

+ fu+gr∂h∂λ

= 0

DhDt

+ h1r∂∂r

(ru) +1r∂v∂λ

⎡ ⎣ ⎢

⎤ ⎦ ⎥= 0

€

g∂h∂r

= f v +v

2

r

€

Ω =v r

Basic state satisfies Gradient wind balance:

Perturbations are functions of (r,λ,t):

Basic state angular velocityLinearized

From Nolan et al (2001)

€

∂un∂t

+ inΩun − ( f + 2Ω)vn + g∂hn∂r

= 0

∂v∂t

+ inΩvn + ( f + Ω +∂vn∂r

)un + ginrhn = 0

∂hn∂t

+ inΩhn + h1r∂∂r

(run ) + hinrvn + un

∂h∂r

= 0

€

[u',v',h'] = [un (r,t),vn (r,t),hn (r, t)]einλ

Basic-State profile• Modified Rankine vortex (α=0.5)• Linear vorticity profile (all the way; to 100km)• Changing aspect ratio (increase h)

Perturbation profile• Balanced Gaussian profile (20% of basic-state)

Case studies

a. From VRW dispersion relations• 2D non-divergent inviscid flow on an f-plane:

• Linearized Shallow-Water equations on an f-plane :

• All parameters are determined by basic-state except for n and k, and n is known once you specify the perturbation. Only k is left for estimation.€

ω =nΩ0 +nRξ0

q0

(∂q0 ∂r)[(k 2 + n2 R2) + γ 0

2]€

ω =nΩ0 +nR

∂ζ 0 ∂r(k 2 + n2 R2)

Abs(qn) Real(qn)

b. From Hovmoller diagramReal(qn*Exp(inθ))

Cgr Cpr Cpλ

1. Modified Rankine Vortex

Choose α=0.5(Same as HW#2)

Find k ---> Time-Radius Real(qn)

€

k = k0 − ntΩ0′ ?

Linear? Approximate rate: 4x10-8

€

−n × Ω0′(r = 50km) = 2.825 ×10−8

Find k ---> Time-Radius Real(qn)

€

k = k0 − ntΩ0′ ?

Linear? Approximate rate: 4x10-8

€

−n × Ω0′(r = 50km) = 2.825 ×10−8

n=2

n=3

Abs(qn) Real(qn) Real(qn*Exp(inθ))

t, k,

t, k,

Cpr

Cgr

n=2 n=3

n=2 n=3

Cpλ:

Counter-clockwiseAbout 20 m/s

Radial wavenumber increases with time:

n=3 is faster than n=2.Azimuthal propagation

slows down faster.

2. Linear Vorticity profile

RMW=75km RMW=75km

Vorticity extend=100km Vorticity extend=100km

2x10-3 4x10-3

• By simplifying (linearizing) the vorticity profile, we hoped to have better approach of the theory.

VRW can be generated by perturbations far from core where almost no gradient exist.

RMW

2*RMW

3*RMW

3. Change Aspect Ratio• To investigate if increasing H (decrease aspect ratio)

can make the dispersion relation approach 2D Hmax = 10 km

ratio= 300km/10km = 30Hmax = 300 km

Ratio = 300km/300km = 1

Identical? Only different in magnitude.

€

ω =nΩ0 +nRξ0

q0

(∂q0 ∂r)[(k 2 + n2 R2) + γ 0

2]

€

ω =nΩ0 +nRξ0

q0

(∂q0 ∂r)[(k 2 + n2 R2) + γ 0

2]

€

q =f +ζh

→∂q∂r

=1h∂ζ∂r

−ζh2

∂h∂r

⇒ (∂q0 ∂r)q0

=1q0h0

∂ζ 0

∂r−ζ 0

q0h02∂h0

∂r

Conclusion and discussion• By perturbing the vortex at some distance (~ 2xRMW or 3xRMW),

VRW can be radiating outward from core. • Perturbations of different azimuthal wavenumber shows consistent

results from theory. • While radial wavenumber is the key to the dispersion relation, it is

also a relatively difficult parameter to determine. In our case studies, k is usually one to two order larger than observation value ( ~ 10-5).

• Cpr matches with the theory pretty well with different vorticity profile. This might be the dominant “apparent” terms in the propagation speed, and so does Cpλ.

• Cgr from theory is two order less than calculated from Hovmoller diagram.

• Changing aspect ratio does not change the vortex evolution because the dominant term of h in potential vorticity gradient cancel with the h in mean potential vorticity.

Conclusion and discussion

• Some thoughts why Cgr is not matching:• dqdr v.s. wavelength• Planetary Rossby Wave: – beta ~ 10-11 m-1s-1; wavelength ~ 106m

• Vortex Rossby Wave: – dqdr ~ 10-12 -10-11 m-1s-1; wavelength 103 - 104m