GEF3450
Exercises for group session:
QG-dynamics and barotropic Rossby Waves
Ada Gjermundsen
E-mail: ada.gjermundsen@geo.uio.no
November 5, 2015
Figure 1: figure from http://www.southamptonweather.co.uk/jetstreamtracker.php
Rossby waves:
While the jet stream generally points roughly due east, it often meanders
north and south. These meanders are the signature of so-called Rossby
waves, which are similar to the familiar waves on the surface of water. However, instead of water moving up and down, Rossby waves consist of large
air masses, often thousands of kilometers across, moving north and south.
1
When an air mass moves north it starts to spin clockwise (when viewed
from above), forming what is called an anticyclonic weather system, a highpressure system associated with settled weather. When air moves south it
spins anti-clockwise and becomes cyclonic, a low pressure system which is
prone to wet weather.
A Rossby wave can therefore lead to a string of alternating high and low
pressure systems, with the jet stream snaking around them from west to
east. Like water waves, Rossby waves generally move relative to an observer on the ground, and this movement leads to changes in the weather
from week to week. In fact, the Rossby Waves themselves always move towards the west, which means they are always swimming upstream against the
eastward-flowing jet. If conditions are right and the wave speed matches that
of the jet, the wave will remain stationary. Then the high- and low-pressure
systems are no longer moving relative to the ground, and a persistent weather
regime is born.
Text from http://www.southamptonweather.co.uk/jetstreamtracker.php
Figure 2: figure from http://www.southamptonweather.co.uk/jetstreamtracker.php
Oceanic Rossby waves are radically different from the surface waves we are
all familiar with (i.e., the ones that enable fun activities like surfing and seasickness). They are huge, stretching for hundreds of kilometers horizontally.
They travel only from east to west across the oceans; open-ocean Rossby
waves are physically incapable of traveling in the opposite direction. They
cause water to meander in wide loops north and south, but only move the
surface of the ocean up or down a few centimeters. They routinely cross entire ocean basins, but take their sweet time doing it: a Rossby wave starting
on the West Coast of the US might take ten years to reach Japan.
2
Figure 3: figure from http://www.nwrfc.noaa.gov/nwrf/papers/Tech_Memo_Nwswr253/finfig14.gif
So who cares? Rossby waves, as it turns out, are important for a variety of
processes on earth. In the ocean, they play a critical role in El Niño, interacting with several other forces to produce the sloshing of warm water back
and forth across the tropical Pacific every two to seven years. Further, they
are one of the major ways that this El Niño signal is transmitted across the
oceans. In the atmosphere, Rossby waves give us most of our weather in the
mid-latitudes. But I just kinda like thinking about them - these huge, silent
disturbances, propagating majestically across the oceans, guided by nothing
but strange effects of the Earth’s shape and rotation. Who knew anything
that weird existed here?
Text from http://www.oceanographerschoice.com/2011/06/rossby-waves/
The Rossby waves described in the text are baroclinic Rossby waves which
we’ll work on later in the course. For now let’s look at the barotropic Rossby
waves.
The barotropic Rossby wave equation:
∂
∂ 2
∂
+U
∇ ψ+β ψ =0
∂t
∂x
∂x
The barotropic Rossby wave dispersion relation:
ω = kU −
3
βk
+ l2
k2
(1)
(2)
Exercises:
Exercises from “Fluid Mechanics” by Kundu and Cohen, “Atmosphere, Ocean and Climate Dynamics” by Marshall and Plumb, “Atmospheric Science” by Wallace and Hobbs
and“Dynamic Meteorology" by Holton and Hakim.
1 Show that the Boussinesq continuity equation:
∂u ∂v ∂w
+
+
=0
∂x ∂y
∂z
can be written as
∂ua ∂va ∂w
+
+
=0
∂x
∂y
∂z
2 In this exercise, using the linearized form of the vorticity equation and
the β-plane approximation, you’ll derive the Rossby wave speed for a
homogeneous incompressible ocean of depth h.
From the QG vorticity equation (i.e. eqn. 4.11 in the old GDF notes
and eqn. 4.2 in the new version of the GFD notes) we have:
Dg ζg
∂u ∂v = −f0
+
− βvg
Dt
∂x ∂y
(3)
Assume a motionless basic state and small perturbations that depend
only on x and t,
u = u0 (x, t)
v = v 0 (x, t)
h = h0 (x, t) + H
where H is the mean depth of the ocean.
i) First, we need the linearized version of the vorticity equation. Use
the expressions for the velocities above and equation 3 to derive the
linearized QG vorticity equation without forcing (i.e. equation 4.31 in
the new GFD notes with U = 0)
ii) Show that the linearized version of the vertically integrated continuity equation for a homogeneous layer
can be written as
∂h
∂
+
(hx) = 0
∂t
∂x
(4)
∂u0
1 ∂h0
=−
∂x
H ∂t
(5)
assuming H >> |h0 |
4
ii) With the aid of equation 5 and the geostrophic wind relation,
0
vg0 = fg0 ∂h
∂x , show that the linearized QG vorticity equation derived in
i) can be written in the form
f 02 0
∂h0
∂ ∂2
−
h
+
β
=0
∂t ∂x2
gH
∂x
(6)
iv ) Show that h0 = h0 eik(x−ct) is a solution provided that
c = −β(k 2 +
f 0 2 −1
)
gH
v ) If the ocean is 4 km deep, what is the Rossby wave speed at latitude
45◦ for a wave of 10.000 km zonal wavelength?
3 Using
u = U + u0
and
u0 = −
∂ 0
ψ
∂y
v0 =
v = v0
∂ 0
ψ
∂x
ζ 0 = ∇2 ψ 0
the linearized form of the QG vorticity equation can be written as
∂
∂ 2 0
∂ψ 0
+U
∇ ψ +β
= −fo ∇H · ~uH
∂t
∂x
∂x
(7)
(i.e. equation 4.31 in the new GFD notes with a forcing term given in
terms of a horizontal divergence field) where ~uH = uı̂ + v̂
i) Suppose the horizontal divergence field is given by
∇H · ~u = A cos[k(x − ct)]
where A is a constant. Find a solution for the corresponding relative
vorticity field.
ii) What is the phase relationship between vorticity and divergence?
iii) For what value of c does the vorticity become infinite?
5
4 i) Derive an expression for the group velocity of a barotropic Rossby
wave with dispersion relation
ω = kU −
βk
k 2 + l2
ii) Show that for stationary waves the group velocity always has an
eastward zonal component relative to earth. Hence, Rossby wave energy propagation must be downstream of topographic sources.
Challenging:
5 In this problem you will derive an equation for the QG streamfunction
of the ageostrophic circulation (in the atmosphere). You should assume
dθ̄
constant f and dz
, and that the geostrophic flow is two dimensional
(y − z) and
ug = ug (z, t)
vg = vg (z, t)
i) Starting with geostrophic and hydrostatic balance
u~g =
1
k̂ × ∇h p
ρ0 f
∂p
ρ0 g
θ
=
∂z
θ̄
show that the maintanance of thermal wind balance requires
Dg ∂ug
g Dg ∂θ
=−
Dt ∂z
θ̄f Dt ∂y
ii) Determine the left-hand side of the result in i) from the ug momentum equation
Dg ug
= f va
Dt
and interpret the result physically.
iii) Determine the right-hand side of the result in i) from the QG
thermodynamic energy equation
Dg θ
dθ̄
= −w
Dt
dz
and interpret the result physically.
6
iv ) Now, use your results from ii) and iii) in i) using a streamfunction
for the ageostrophic wind:
va =
∂
ψ
∂x
w=−
∂
ψ
∂y
express your result with all streamfunction terms on the left-hand side
of the equation.
v ) Assume that the right-hand side of your result in iv ) reaches a local
minimum in the midtroposphere. Sketch the ageostrophic circulation
streamlines in a (y − z) cross-section along with arrows showing va and
w.
Solutions:
1
2 v ) c = −24.3 ms−1
f0 Ak
3 i) ζ 0 = ∇2 ψ 0 = − [(c−U
sin(k[x − ct])
)k2 +β]
ii) For (c − U ) > − βk : vorticity leads divergence by π2
For (c − U ) < − βk : vorticity lags divergence by π2
iii) we have that ∇2 ψ → ∞ for (c − U )k 2 + β = 0 or c = U − kβ2 , which
is the free Rossby wave speed
4 i) Ugroup =
∂ω
∂k
=U+
∂ω
∂l
=
2βl
(k2 +l2 )2
Vgroup =
β(k2 −l2 )
(k2 +l2 )2
ii) For stationary waves, ω = 0 or U =
2 −l2 )
β(k2 +l2 )
+ β(k
(k2 +l2 )2
(k2 +l2 )2
2
Ugroup = − (k2βk
2 +l2 )2 > 0
and Ugroup =
Thus
5
7
β
(k2 +l2 )