Small Rossby Number Corrections to Shallow Water Quasigeostrophy
Alana McKenzie, Colin Macdonald & David J Muraki
Simon Fraser University
Why does rotating shallow water display so
little asymmetry in the evolution of balanced
vortices at small Rossby number?
Vortex Dynamics in swQG
Shallow Water Potentials: Version II
• small Rossby number perturbation solution
• PV dynamics with ∇8 hyperdiffusion
• towards a unifying view for QG & gravity waves . . .
H ∼ H0 + R H1 + . . .
Dq
= −ν ∇8q
Dt
RF1 + ...
F ∼
In contrast, there is a marked asymmetry of
surface QG vortices in a stratified fluid.
+1
Small Rossby Number Corrections to QG
+
φx + χy
−u = ψy
+
φy − χx
• PV & vorticity evolution with R = 0.1
1
G ∼
v = ψx
RG + ...
B 2
+ ∇φ
Q
h = ψ
• next-order balanced inversions
2
0
∇ H −B
−1
−1
H
0
• an exact PV streamfunction (q = 0 ⇒ ψ = 0)
= q
Shallow Water Equations
∇2 F 1 − B
• single-layer, f -plane rotation
∇2 G1 − B −1 G1 = − J(H 0, Hy0)
Du
R
−v
=
Dt
Dv
R
+u
=
Dt
Dh
+ (B + R h)(ux + vy ) =
R
Dt
∇2ψ − B −1Q ψ = q
F 1 = − J(H 0, Hx0)
• inversions via ageostrophic vorticity & divergence
∇2 H 1 − B −1 H 1 = q H 0
− hx
!
γ =
• PV dynamics by next-order (swQG+1) winds
− hy
• pseudo-spectral computation with Fourier inversions
Dq
= q t + (u0 + R u1)q x + (v 0 + R v 1)q y = 0
Dt
0
;
=
∇2 χ
Q
∇ − B ((B + R h)δ)
2
= R −u(∇2h)x − v(∇2h)y + ∇2(uhx + vhy )
• Polvani, McWilliams, Spall & Ford (1994)
Dδ
R
Dt
• near-symmetric dynamics from initial balance
1 + R (vx − uy )
Q = 1 + Rq =
1 + R B −1 h
n
Freely-Decaying Vortex Dynamics
• potential vorticity
= ux + vy
• nonlinear wave equations for γ, δ
Dγ
R
−
Dt
gH
B =
(f L)2
• vorticity evolution in rotating SW with R = 0.05
− γ
n
= R 2J(u, v) − δ 2
• advection of disturbance PV
Dq
= 0
Dt
• local convergence/divergence can change mean(PV)
Gravity Wave Dynamics
• linearized wave equations (Q=1)
R γt − B ∇2δ + δ
• divergence at O(R)
Shallow Water Potentials: Version I
• a simple extension to quasigeostrophic thinking . . .
−
−u = Hy
h = H
(∇2 − B −1) (u1x + vy1) = −J(H 0, q)
• Hakim, Muraki & Snyder (2002)
G
• advection by divergent winds, yet mean(PV) = 0!
• asymmetric balanced dynamics with R = 0.1
+
F
• irreversible cooling of lower surface
− B Gx + B Fy
• surface potential temperature evolution in sQG
R δt
ZZ
+1
(ux + vy ) q ∼ R
ZZ
(∇2 − B −1) (u1x + vy1) H 0 = 0
Downloaded 19 Nov 2004 to 128.59.51.151. Redistribution subject to AIP license or copyright, see http://chaos.aip.org/chaos/copyright.jsp
• inversion equations for H, F, G
2
∇ H −B
2
∇ F −B
2
∇ G−B
−1
−1
−1
H = q + Rh
F =
G =
R
+
B
o
Divergent PV Flow, Yet Mean Conserving?
q t + (uq)x + (vq)y − (ux + vy ) q = 0
v = Hx
= v x − uy − ∇ 2 h
• near-symmetry consistent with Polvani, et.al.
• Rossby & Burger numbers
U
R =
fL
δ
−∇
B 2
∇ φ−φ
Q
2
(

R
Dh
+ h(ux + vy )
Dt
!
)
!

Du 
Dh
+ 
+ h(ux + vy )
B
Dt
Dv
−
Dt
x
+
y
Dt 
Oddly, the divergent swQG+1 dynamics still
conserves mean PV.
This is a significant restriction on the degree
of asymmetry which can develop at O(R).
= 0
− γ = 0
• dispersion relation
R2ω 2 = B (k 2 + l2) + 1
• exact nonlinear fast manifold with q=0
Du
R
−v =
Dt
Dv
R
+u =
Dt
− B (vx − uy )x
− B (vx − uy )y
Gravity Wave Generation by PV Dynamics
• PV forcing terms for wave generation
o