Shallow-water vortex quasi-equilibria
and their stability
Hanna Płotka & David G. Dritschel
Vortex Dynamics Group, School of Mathematics and Statistics, University of St Andrews
Steadiness parameter ǫb
Motivation
We define the dimensionless steadiness
q H parameter ǫb to be
1
2 ds
(p
−
p̄)
L C
ǫb =
,
p̄
H
H
where L = C ds is the arc length of the PV contour, and p̄ = (1/L) C p ds is the
mean Bernoulli pressure around the PV contour.
1000
3000
t
εb
0.03
0.3
0.01
500
1000
−3
.5
−4
−5
5
.2
−0
−2
−0
500
1500
t
−0.1
5
15
−0.
.2
1500
−0
.2
−0.1
−0
.2
5
−2
−1.5
−1.5
2500
t
−2
−2
.2
.5
−0
−0.1
−0.1
.15
−0
.5
−4
−4
.5
5
.
−3
5000
t
−3
−3
−4
−5
0.05
0.5
1000 1500 2000
−2
.5
500
0.1
0.0015
0.005
εb
0.010
εb
0.0025
εb
0.0035
0.015
0.07
It is beneficial to examine geophysical fluid motions through the concept of balance,
in which the flow can be decomposed into a dominant, ‘balanced’ component of
motion relating to spinning vortices, and an ‘unbalanced’ component relating to the
propagation of inertia-gravity waves (IGW’s).
−0
.1
5
−0
.2
−0.05
−1.5
−0.05
−1
Examples of geophysical vortices. Left: Gulf stream rings; Right: The Great Red Spot on Jupiter (as
seen from Voyager 2 in 1979, source MPI/Hulton Archive/Getty Images).
−0.3
−0.45
−0.3 −
0.
−0.45 25
−1
−1.5
−0
.35
Quasi-geostrophic equilibria
−0.4
−0.2
−3
−0.1
5
−0.2
−0.4 −0.
35
−0.5
0
0
−0.15
−0.4 −0
.35
−0.5
0
−0.45
−2
−0
−0.45
.2
5
−0.3
−2
−0.25
−1
−0.05
−4
.5
−1.5
−5
−4
−2.5
−3.
5
−0.1
−0.1
−3
−3
.5
−3
−4
−0.05
−1
−0
.2
−5
5
−0
−0.1
.2
5
−0.1
5
.2
−0
−0
−2
25
.
.5
−2
−1.5
−2
−1.5
−0
.5
−2
.2
−0
.15
−0.1
−0.1
.15
−0
.2
−0
Examples of different types of behaviour of the SW quasi-equilibria. Here, we see the cases when the
state is stable (γ = 1, R = 0.3, λ = 0.190; column 1), vacillates (γ = 3, R = 0.2, λ = 0.015; column 2),
splits asymmetrically (γ = 1, R = 0.3, λ = 0.185; column 3), and splits symmetrically (γ = 3, R = 0.2,
λ = 0.010; column 4). The top row shows the value of the stability parameter ǫb through time; the red line
indicates the time at which we show a snapshot of the quasi-equilibrium in the bottom row. The bottom
row shows the potential vorticity contours (in red) and Bernoulli function contours (in black) at times
t = 2356, t = 532, t = 318 and t = 318, respectively. Here, |x|, |y| ≤ π.
0.30
0.35
We use the quasi-geostrophic shallow-water
(QGSW) model to generate two-fold symmetric
vortex patch relative equilibria (or just ‘equilibria’).
We use the simplest form of the model, with no
forcing, no damping, no topography and constant
planetary vorticity f . The QGSW model consists
of a single ‘prognostic’ equation for the material
(conservative) advection of QGSW potential vorticity (PV) q,
Dq ∂q
∂q
∂q
=
+u
+v
= 0,
Dt
∂t
∂x
∂y
and a Helmholtz-type ‘inversion relation’ providing the (non-divergent) flow field u from q,
γ
∂ψ
∂ψ
2
−2
; v=
, Growth rates σr of modes 1 and 2 in
(∇ − LD )ψ = q ; u = −
∂y
∂x
the γ–λ parameter plane. The contour
where ψ is the streamfunction and LD = c/f is the interval is 0.01.
Rossby deformation length, with c the short-scale gravity wave speed. The equilibria are spanned by two parameters: γ = L/LD (with L = 1 by imposing constant
area) and the aspect ratio λ. Inside the patch q = 1, while outside q = 0.
−0 −0.5
.35
−0.4
−2.5
−2.5
.5
−4
Vortices are an omnipresent feature of the Earth’s oceans and atmosphere, as well
as the atmospheres of the giant gas planets. To better understand them, we first
examine them through the quasi-geostrophic shallow-water (QGSW) approximation,
which filters out the unbalanced component, and then add further complexity by
solving the full shallow-water (SW) system, which allows IGWs.
−0.5
0
−0.5
0.25
Cyclone-anticyclone asymmetry
0.35
0.40
0.45
0.55
0.5
0.0
0.5
0.50
γ=6
−0.5
10
Rossby number
8
0.0
6
γ=2
−0.5
4
Rossby number
2
−0.5
0
γ = 0.25
0.0
0.00
Rossby number
0.5
0.05
0.0
0.10
0.15
λ
0.20
The ageostrophic component breaks the symmetry between cyclonic and
anticyclonic motions (motions spinning in opposite directions; in the QGSW model
these motions are symmetric).
0.05
0.10
λ
0.15
0.20
0.05
0.10
λ
0.15
0.20
λ
We see: • - the boundary of stability, ◦ - the boundary between vacillations and instability, and N - the
QGSW boundary of stability. The states to the left of the symbols are unstable, while those to the right
are stable (except in the case of vacillations, where there may be a region of vacillating instability).
Optimal PV balance
Selected equilibrium contour shapes for γ = 0.5 (left), 3.0 (middle), and 8.0 (right). In each frame, we
show the equilibrium contours for λ = 0.5, for the largest stable aspect ratio λc , and for the smallest
aspect ratio attainable λf . The plot window shows |x| ≤ 2.2, |y| ≤ 0.8.
−4
Vortex Dynamics Group - University of St Andrews - UK
−6
−10
−12
−10
−12
400
600
200
400
600
200
t
600
t
−4
400
600
t
200
400
t
600
−10
−8
log(v’)
−12
−10
−12
−10
−12
200
−8
log(u’)
−8
log(h’)
0.0008
−6
−6
−6
−4
0.0012
400
−4
t
t
−8
log(v’)
−8
log(u’)
200
600
0.0004
The shallow-water quasi-equilibria are not strictly steady, but radiate such weak
gravity waves that they are steady for all practical purposes. We examine their
steadiness by looking at the extent to which the contours of the Bernoulli function
coincide with those of the PV. For rotating shallow-water flows, the Bernoulli function
takes the form:
1
1
2
2
p = hc + |u rot | − Ωf (x 2 + y 2),
2
2
with u rot = (u + Ωy , v − Ωx) and Ω the rotation rate. In a steady flow, p is constant on
PV contours, for an appropriate choice of Ω.
−6
−4
−10
−12
0.0000
400
0.0000
Steadiness and the Bernoulli function
−8
log(h’)
−6
0.0010
0.0005
δ
200
δ
Using the QGSW equilibria, we introduce ageostrophic effects, including the
generation and propagation of IGWs, in our model and generate ‘quasi-equilibria’ by
solving the full shallow-water equations:
Du
Dv
∂h
2 ∂h
2 ∂h
− fv = −c
,
+ fu = −c
and
+ ∇ · (uh) = 0,
Dt
∂x
Dt
∂y
∂t
where u = (u(x, y , t), v (x, y , t)) is the (horizontal) velocity and h(x, y , t) is the fluid
depth (or height) scaled on the mean fluid depth H.
We use the contour advective semi-Lagrangian (CASL) algorithm, in which we keep
track of contours of constant potential vorticity, and velocity and pressure on a grid 1.
In a procedure called ‘dynamic PV initialisation’ 2 we artificially ramp up the PV
anomaly on every fluid particle to a finite Rossby number R while integrating the full
equations. Here, R is the only additional parameter needing to be introduced.
0.0015
Shallow-water quasi-equilibria
−4
The ‘optimal potential vorticity’ balance method 3, which exploits the property that PV
cannot be transferred into IGWs, has been applied in order to quantify the amount of
unbalanced motion. In this method, we are able to calculate the ‘balanced’ fields, and
by subtracting them from the original fields, we are able to find the levels of IGW
activity. We find that even during the onset of instability, only negligible levels of IGWs
are produced.
200
400
600
t
200
400
600
t
The spatial average of the balanced (in black) and unbalanced (in red) fields for divergence δ and the log
of anomalies in height h′ and velocities u ′ and v ′ obtained from OPV balance. We see two aspect ratios
for γ = 6, R = 0.3: unstable λ = 0.025 (row 1) and stable λ = 0.135 (row 2).
References
1. Dritschel, D. and Ambaum, M. Quart. J. Roy. Meteorol. Soc. 123, 1097–1130 (1997).
2. Viúdez, A. and Dritschel, D. J. Phys. Oceanogr. 34 (2004).
3. Viúdez, A. and Dritschel, D. G. J. Fluid Mech. 521 (2004).
[email protected]
http://www-vortex.mcs.st-and.ac.uk/˜hanna