3 IntroductionJule 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 approximationJohn DeVore
4 IntroductionCharney and DeVore examined the concept of multiple flow equilibria in the atmosphere and blockingCDV 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 perturbationsModel 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)
5 IntroductionSolutions 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.
6 CDV derivationThe 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 + yThe 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 conditionsNo normal transport at the boundaries requires PHI to be constant at y= 0,Ly
7 CDV Derivation The equation used in the model is the QGPV equation 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.
8 CDV Derivation Laplace equation Break the problem into four problems with each having one homogeneous conditionSeparate the variablesSolve x dependent equation and y dependent equation.Use boundary conditions and orthogonality to find coefficients
9 CDV derivation Orthogonality Whenever 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
10 CDV DerivationThe process is similar in the derivation of the CDV modelFirst 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).
11 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
12 CDV modelAs 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 equilibriaThis is a drawback of the CDV model because there is no way to transition between the two stable equilibra points.