Low Frequency Variability and
Climate Regimes:
A look at the Charney DeVore Model
Josh Griffin and Marcus Williams
Outline
 Brief History
 Introduction
 The CDV model
 From Holton
 From Charney
 Examples
 Stochastic forcing
What are we talking about?
 We know that the climate is described as a basic
state flow that is modified by eddy fluxes --needs
better wording-- Low frequency variability describes these eddy
fluxes that last on time scales longer than those of
transient eddies
 Can be anywhere from 7-10 days to interannular
variability
 These persistent anomalies can lead to climate
regimes in the general circulation of the
atmosphere
 These regimes are characterized be either a highindex or low-index state
 Define climate regime
 Graphic as an example
What is the purpose of the CDV
model?
 What does it look at? An extremely simplified
model of a barotropic atmosphere.
 What does it hope to solve? Hoped to describe the
persistance of large amplitude flow anomalies like
blocking or recurring regional weather patterns
 What does it tell us? Examines the equilibrium
mean states of the atmosphere when a damped
topographic Rossby wave interacts with the zonalmean flow. --direct quote from holton
 Model tells us that there are multiple equilibrium
solutions for the atmosphere.
 Both solutions are stable, however only one is
seen. ???
 Charney
Who invented it?
 Hard to find something he didn’t do…
 His PhD thesis took up an entire journal in october
1947
 Important for 2 reasons --list them
 Developed quasi-geostrophic approximations
 Helped proved the concept of numerical weather
prediction was feasible and practical
 Helped come up with concept of barotropic instability
??? (“True” according to wikipedia)
 Helped explain formation of mid-lat cyclones
 Dishpan experiment
 DeVore
 One hit wonder…
 This is his only paper listed on the AMS website
 Apparently works for a company named Visidyne
Let’s talk about models…
 Will be looking at two approaches
 First approach is from Holton
 Is a more ad hoc approach
 Less dynamical than the original CDV paper
 Actually feasible for us to derive…
 Second approach is from CDV Paper
 More dynamical
 Mathematically complex
Holton’s approach
 Start with the barotropic potential vorticity
equation


DhT
H  V   g  f   f 0
t

Dt
 Explain terms
 Why use this equation? It is the simplest model of

topographic Rossby waves
 Make the assumption that the upper boundary is fixed at a
height H and the lower boundary is variable height ht(x,y)
where ht <<H
Now what?


 
f 0 DhT
  u  v  g  f  
t
x
v 
H Dt
 First step is to linearize

u  u  u
v  v  v 
    

v u

x y
 Next we make some assumptions


 flow
v 0
 Zonal
mean
 Take the zonal average



0
x
And then…
 We then integrate the equation w.r.t y

u
f
 v  0 v hT
t
H
 By adding some “forcing??” terms, you
arrive at the equation
 u
t
 
 D u   u  U e

  
 Where

 This
is defined as the barotropic momentum
equation
D u  v 

f0
vhT
H
Now that we have an equation
 The barotropic momentum equation is
dominated by two terms
 D(u) describes the forcing interaction between the
waves and the mean flow
 -kappa(u-Ue) describes a linear relaxation toward an
externally determined basic state flow, Ue
 Since we know D(u), we can plot the
solution if we make some assumptions.
assumptions
 Assume the streamfuction is composed of a
single harmonic wave in the x and y
direction.
 Doing this results in:
x, y  Re0 expikxcosly
hT  Reh0 expikxcosly
where 0 


 We know that


v
x
and
 2  2
 2 2

x
y
HK
f 0 h0
2
 K s  i
2

Plug and chug
 After plugging the wave solutions, D(u) simplifies
 The eddy vorticity flux goes away
 The second term, the form drag, is all that remains
 Explain terms

rK 2 f 02  h02 cos2 ly 
D u  
2 
2
 2uH  K 2  K S2    2



Graphical solution
 Explain the equilibrium points
 Why is one low-index and one high index?
Transition slide into CDV paper
CDV derivation
 The CDV model comprises a Rossby wave mode and uniform zonal
flow over a mountain in a  plane channel.
 The coriolis parameter f is approximated by
 The flow is restricted by lateral walls with width 0< y<Lx and length
0<x<Lx.
 The flow is also periodic in longitude so
 No normal transport at the boundaries requires  to be constant at y=
0,Ly
CDV Derivation

The equation used in the model is the QGPV equation
,  *,

To derive the low order spectral model you must expand
orthonormal eigenfunctions of the Leplace operator.

This derivation is very complex. I will show a more general representation by solving
Leplace’s equation on a rectangle and introducing
 the concept of orthogonality.
and h(x,y) into
CDV Derivation
 Laplace equation
2 u 2 u
u  2  2
x
y
 Break the problem into four problems with each having one
homogeneous condition u  u1  u2  u3  u4

 Next assume that u is a function of a product of x and y
u( x, y)  h( x) ( y)
 Separate the variable to get an ODE for x and y and set equal to
an arbitrary constant.
CDV Derivation
 Solve x dependent equation and y dependent equation. The equation with
two homogeneous boundary conditions will provide you with your
eigenvalues.
d 2h
 h( x )
2
dx
h( L)  0
d 2
 
2
dy
 (0)  0
 (H )  0
 Use boundary conditions to solve for the eigenfunction and orthogonality to
solve for the inhomogeneous initial condition
CDV derivation
 Orthogonality
 Whenever 
L
A(x)B(x)dx  0
it is said that functions are
orthogonal over the interval 0<x<L. The term is borrowed
from perpendicular vectors because the integral is analogous

to a zero
dot product
0
0
nx
mx
sin
dx


0 L
L /2
L

0
L
nx
mx
cos
cos
dx


0
L
L
L /2
L
L
sin
m n
m n
m n
m n  0
m n  0
CDV Derivation
 The process is similar in the derivation of the CDV model
 First you have to non-dimensionalize the QGPV equation.(A1,A2)
 2
h
D


    2  J ,  2   f o  y    f o E  2   *
t
H
2H






 Make the rigid lid approximation   0 and use the characteristic height, the
timescale, the horizintal length scale, and the characteristic amplitude of the
topography.
2
 The non-dimensionalized QGPV becomes


 2  J  ,  2   J  , h   
 C 2   * : A1
t
x
f h
f D

  0 0 , 
,C  0 E
H
k
2H




CDV Derivation

Represent h(x,y) and  in terms of sines and cosines(A4).
*
1
y
 2  co s(x ) sin 


2
b 
*
y
1  2 co s


0
b 
h ( x, y ) 


*


Expand  into three orthonormal modes(A3).
   1  2  3
 y

b
 
 1   11   1 2 cos
 y

b
 
 y
  33   3 2 sin(x ) sin  
b
 2   22   2 2 cos x  sin 
3
CDV derivation
 Insert A3 and A4 back into the A1 and utilize the orthonomality of the
eigenfunctions  and let xi  b , .  f0h0 H
i
 This leads to the following equations known as the CDV equations(A5).
.
x
1

 b x3  C x1  x1
*

1


  a b x1    x3  Cx2
2 

.
1
1


x 3  a b x1  2   x2  2 a x1  Cx3
with
2b
3
3
a 
,  
, C 
C
2
1 b
4 2
4 2
.
x
2
 These equations define the low-order spectral model.The CDV equations
are solved to find the equilibrium points
CDV model
 As we found from holton, the system has three equilibrium point. One
unstable and two stable(Show graphic again?)
 For arbitrary initial conditions the phase space trajectories always tend to one
of the two stable equilibrium
 This is a drawback of the CDV model because there is no way to transition
between the two stable equilibrium points.
CDV model
Example of a blocking climate regime in mid-lattitudes
Stochastic slide 1
•As was shown earlier, there is no way to start a
transition from one stable equlibria to another
•Papers by Eggert (1981) and Sura (2002) discuss
this transition between equilibrium through
stochastic processes
Stochastic slide 2
 Obviously, since the points are equilibrium points,
the solutions tend to go to one of those points and
remain there in the CDV model
 By adding the stochastic white noise to the
system, it generates a mechanism by which the
system can switch between the equilibrium points
Matlab examples
 Holton provides an example of a twomeridional-mode version of the CharneyDeVore model
 Now we’ll show a few examples of how
topographic forcing alters the
streamfunctions, both in structure and
persistence.