Low Frequency Variability and Climate Regimes: A look at the Charney-DeVore Model

1. Low Frequency Variability and Climate Regimes:A look at the Charney-DeVore Model Josh Griffin and Marcus Williams

2. Outline • Brief History • Introduction • The CDV model • From Holton • From Charney • Examples • Stochastic forcing

3. What are we talking about? • We know that the climate is described as a basic state flow that is modified by eddy fluxes • 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 interannual variability • These persistent anomalies can lead to climate regimes in the general circulation of the atmosphere • These regimes are characterized be either a high-index or low-index state • High index state has a strong zonal flow and weak waves • Low-index state has weak zonal flow and high amplitude waves

4. What is the purpose of the CDV model? • The CDV model looks at a simplified version of a barotropic atmosphere. • Goal of the model is to describe the persistence of large amplitude flow anomalies like blocking or recurring regional weather patterns • Does this by examining how topographically forced Rossby waves interact with the zonal mean flow • Model shows that there are multiple equilibrium states for the atmosphere • There are two stable equilibrium points and one unstable • The stable points consist of one low-index and one high-index state

5. About the authors • Jule Charney • Hard to find something he didn’t do… • His PhD thesis took up an entire journal in October 1947 • Emphasized importance of long waves in the upper atmosphere on the entire atmosphere • Provided a simplified way to examine perturbations along the waves • Developed quasi-geostrophic approximations for planetary-waves • Helped prove the concept of numerical weather prediction was feasible and practical • John DeVore • One hit wonder… • This is his only paper listed on the AMS website • Apparently works for a company named Visidyne

6. About the model • Will be looking at two approaches • Holton’s Approach • Is a more ad hoc approach • Less dynamical than the original CDV paper • Actually feasible for us to derive… • Original CDV Paper • More dynamical • Mathematically complex

7. Holton’s approach • Start with the barotropic potential vorticity equation • 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

8. Now what? • First step is to linearize • Next we make some assumptions • Zonal mean flow • Take the zonal average

9. And then… • We then integrate the equation with respect to y • By adding some external forcing terms, you arrive at the equation • This is defined as the barotropic momentum equation

10. 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 • Eddy vorticity flux • Surface pressure torque term • - (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.

11. More Assumptions • Assume the streamfuction is composed of a single harmonic wave in the x and y direction. • Doing this results in: • We know that

12. 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

13. Graphical solution • Explain the equilibrium points • Why is one low-index and one high index?

14. 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

15. Topography in the -plane

16. 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.

17. CDV Derivation • Laplace equation • Break the problem into four problems with each having one homogeneous condition • Next assume that u is a function of a product of x and y • Separate the variable to get an ODE for x and y and set equal to an arbitrary constant.

18. CDV Derivation • Solve x dependent equation and y dependent equation. The equation with two homogeneous boundary conditions will provide you with your eigenvalues. • Use boundary conditions to solve for the eigenfunction and orthogonality to solve for the inhomogeneous initial condition

19. 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

20. CDV Derivation • The process is similar in the derivation of the CDV model • First you have to non-dimensionalize the QGPV equation.(A1,A2) • Make the rigid lid approximation and use the characteristic height, the timescale, the horizontal length scale, and the characteristic amplitude of the topography. • The non-dimensionalized QGPV becomes

21. CDV Derivation • Represent h(x,y) and in terms of sines and cosines(A4). • Expand into three orthonormal modes(A3).

22. CDV Derivation • Insert A3 and A4 back into the A1 and utilize the orthonomality of the eigenfunctions and let . • This leads to the following equations known as the CDV equations(A5). • These equations define the low-order spectral model.The CDV equations are solved to find the equilibrium points

23. 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.

24. CDV Derivation • Recall from class notes that the previous system of equations can be represented as • We want to find the roots of so we can determine the steady states of the dynamical system • The roots are where

25. CDV Derivation • Use taylor expansion to perturb the equilibrium through a small distance • Linearize the equation and introduce the Jacobian Matrix • Where

26. CDV Derivation • The change in the perturbation over time is governed by the term • Finding the eigenvalues() of the Jacobian matrix at the equilibrium yields the stability • If all eigenvalues have a negative real part the equilibrium is stable. If just one eigenvalue has a positive real part then the equilibrium is unstable.

27. CDV Model Example of a blocking climate regime over the Pacific Ocean

28. Examples

29. Examples

30. Stochastic Forcing • As was stated earlier, there is no way to start a transition from one stable equilibrium to another in the CDV model • Papers by Eggert (1981) and Sura (2002) discuss ways to transition from one equilibrium point to another through stochastic noise

31. Stochastic Forcing • In the CDV model, the solutions tend to go to one of the two stable equilibrium points and remain there. • By introducing the stochastic white noise to the system, it generates a mechanism by which the system can switch between the equilibrium points • Adding noise helps to kick the “marble” from one state to another • This is method is similar to to notes from week 11 and 12 • In that section, we discussed how adding Gaussian white-noise helps understand climate variability • However, Sura (2002) also discusses the impact of multiplicative white noise on the system

32. Matlab example • For anyone who is interested, Holton provides a Matlab script for the CDV model that contains two meridional wave modes. • Applies to question M10.2 in the 4th edition • By alternating the forcing of the zonal flow and the amplitude of the zonal flow, can achieve different climate regimes similar to the single wave CDV model • Through the addition of the second wave, oscillation is seen in the structure of the stream functions, but transitions from one regime to another are still not seen

33. References • Charney • Sura • Egger • Holton