Transcript Charney DeVore model

Charney DeVore model
Josh Griffin and Marcus Williams
Outline
Introduction
• Jule Charney is a well known scientist who has made multiple
contributions to NWP including the Charney model.
• Charney developed a set of equations for calculating the large-scale
motions of planetary-scale waves known as the "quasi-geostrophic
approximation
• John DeVore
Introduction
• Charney and DeVore examined the concept of multiple flow
equilibria in the atmosphere and blocking
• CDV analyzed a simple model of a barotropic atmosphere in which
an externally forced zonal flow interacts non-linearly with topography
and with externally forced wave perturbations
• Model hoped to describe the persistence of large amplitude flow
anomalies like blocking or the recurring regional weather patterns.
• Since the motions are large scale, they will be quasi-geostrophic
and governed by the conservation of potential vorticity(PV)
Introduction
• Solutions to model yield two stable equilibria points and one
unstable, transitional equilibria point.
• The first stable equilibrium point is characterized as low-index,
strong wave component with weak zonal flow (blocking)
• The second stable equilibrium point is characterized as high-index,
weak wave component with strong zonal flow.
• There are several ways the model can be derived but we will focus
on two derivations in particular.
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 f = f +  y
• The flow is resticted by lateral walls with width 0< y<Lx and length
0<x<Lx.
• The flow is also periodic in longitude so  (x,y,t)= (x+Lx,y,t)
Boundary conditions
• No normal transport at the boundaries requires PHI to be constant
at y= 0,Ly
CDV Derivation
• The equation used in the model is the QGPV equation
 2
h
D


    2  J ,  2   f o  y    f o E  2   *
t
H
2H






• To derive the low order spectral model you must expand  , ,
and h(x,y) into 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.
CDV Derivation
• Laplace equation
 2u
 2u
u 

x 2
y 2
• Break the problem into four problems with each
having one homogeneous condition
• Separate the variables
• Solve x dependent equation and y dependent
equation.
• Use boundary conditions and orthogonality to find
coefficients
CDV derivation
• Orthogonality
– Whenever 0 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
L
•
n x
m x
 0 mn
sin
dx


0
L
L
L / 2 m  n
L
n x
m x
0 mn

cos
cos
dx


0
L
L
L / 2 m  n  0
L
sin
CDV Derivation
• The process is similar in the derivation of
the CDV model
• First you have to non-dimensionalize the
QGPV equation.(A1,A2)
• Represent h(x,y) and PHI* in terms of
sines and cosines(A3), and expand PHI
into three orthonormal modes(A4).
CDV derivation
• Insert A3 and A4 back into the A1.
• This leads to the following equations
known as the CDV equations.
• The CDV equations are solved to find the
equilibrium points
CDV model
• As we found from holton, the system has
three equilibria 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 equilibria
• This is a drawback of the CDV model
because there is no way to transition
between the two stable equilibra points.
CDV model