Doctoral Thesis Proposal:
Looking for Coupled Midlatitude Climate Variability using Adjoint and Eigensystem Techniques

Jason Goodman
Program in Atmospheres, Oceans and Climate
Department of Earth, Atmospheric and Planetary Sciences
Massachusetts Institute of Technology

1. Introduction

This thesis proposal is a plan of action toward improving our understanding the internal variability of the climate system on timescales from 5 to 50 years. Changes on these timescales can have adverse societal impacts; they also tend to obscure long-term climate change, making detection of anthropogenic influence more difficult.

The most pronounced patterns of atmospheric variability in the Northern midlatitudes are the North Atlantic Oscillation (NAO) and the Pacific-North America pattern (PNA). Each pattern explains 30-40% of the sea-level pressure (SLP) variability over its basin on timescales longer than a month (Cayan, 1992). An index of the NAO defined by SLP difference between Iceland and the Azores or Portugal during winter months shows large decadal and interdecadal variability. The PNA appears to show similar patterns. Several authors (Kushnir, 1994, Deser & Blackmon, 1993) demonstrate a connection between the atmospheric NAO pattern and sea-surface temperature patterns in the North Atlantic. However, the nature of this connection is currently unknown. There are three ways the atmosphere and ocean can interact: On sub-annual timescales, the atmosphere's intrinsic variability forces an ocean response. On centennial timescales, there is evidence that oceanic changes force the atmosphere. Between these regimes, there may or may not be a mutual coupled interaction in which variability arises as a fundamentally coupled process.

One way to find out whether active atmosphere-ocean coupling can lead to decadal variability is to look for coupled growing modes within the system. If a model of the atmosphere and ocean can support growing coupled modes with interannual or interdecadal period, it may help us understand the observed variability in the same way that the simple Eady and Charney models of baroclinic instability help to explain the existence of synoptic-scale weather systems. In Goodman and Marshall (Journal of Climate; to appear, 1998), we discuss a simple coupled model that supports such coupled modes; a similar model was studied by Qiu and Jin (1997) to explain the Antarctic Circumpolar Wave.

A second way to find out whether the atmosphere and ocean undergo coupled interaction is by studying the response of one system to forcing by the other. Numerous studies of the steady-state atmospheric response to thermal forcing have been performed (Kushnir and Held, 1996; Latif et al., 1996, Palmer and Sun, 1985) to find out whether an atmospheric model will produce NAO- or PNA-like pressure anomalies when forced with the sea-surface temperature patterns observed to covary with the NAO or PNA. The results and interpretations of these experiments differ wildly between authors; so far, there is very little consistency between the various models.

I propose to carry out a series of experiments, using a hierarchy of models of increasing complexity, which will hopefully clarify the coupling connection between atmosphere and ocean in midlatitudes. Since this work builds and elaborates upon work discussed in Goodman and Marshall, and upon work actively underway within the climate program at MIT, a brief review of this research is in order.

2. Foundations

Over the past year and a half, Dr. Marshall and I have been working on a simple, analytically-soluble model of atmosphere-ocean interaction in middle latitudes. The model couples a time-evolving 1 1/2-layer quasigeostrophic ocean without boundaries to a steady-state 2-layer QG atmosphere with constant zonal winds. The atmosphere is forced by relaxation to an equilibration temperature determined by SST: the ocean is forced by surface windstress. SST is determined by a mixed-layer equation forced by air-sea heat fluxes, entrainment fluxes, and advection.

For parameters typical of the North Atlantic, this model exhibits decadally-oscillating coupled modes which grow exponentially with time; the wavelengths and timescales resemble the NAO pattern. The wave oscillates decadally due to Rossby wave propagation in the model ocean: it grows exponentially due to a positive feedback between the ocean's response to winstress forcing and the atmosphere's response to thermal forcing. Such a mechanism could explain the pronounced decadal coupled variability observed.

Meanwhile, Dr. Marshall's group at MIT has been developing a coupled model of intermediate complexity by converting the MIT parallel ocean model into a coupled atmosphere-ocean model. The intent here is to create a tool to investigate the role of dynamics in climate variability, using forcing terms of intermediate complexity.

One attractive feature of this particular climate model is that it will come with an adjoint model, generated using the TAMC automatic tangent linear / adjoint compiler developed by Ralf Giering. Using this tool, changes to the model can be incorporated into the automatically-generated adjoint with no additional effort. This will enable me to carry out the maximum-excitation experiment described below.

3. Proposed Experiments

   
3.1 Coupled Growing Modes in Complicated Geometry

The Goodman and Marshall (1998) model demonstrates a mechanism which can produce growing coupled modes which exhibit decadal variability. However, the model is an extremely simple one, in both its dynamics and its geometry. The model has uniform zonal winds, no ocean currents, and no land-sea boundaries. The existence of land boundaries and mean ocean currents might disrupt the coupled modes, and wavy background flow in the atmosphere could significantly modify the atmosphere's response to thermal forcing.

On the other hand, nonuniform winds are capable of trapping planetary waves: Marshall and Molteni (1993) found nearly-stationary atmospheric ``neutral vectors'', some of which resemble the NAO and PNA patterns. Interestingly, we find that oceanic forcing must excite nearly-stationary free atmospheric waves in order for coupled growth to occur. It's possible that by trapping planetary waves, coupled growth could occur quite easily via the mechanism described in our paper.

To test these ideas, I will take the physical system described in Goodman and Marshall (1998) and place it in a more realistic geometry, and then look for growing coupled modes within this system. For nonuniform background and boundary conditions, the resulting equations are linear but not constant-coefficient. They may be discretized to form a large set of ODEs:

$$\frac{\partial}{\partial t} \mathbf{A} \left(\begin{array}{c}... ...atm}} \\ \mbox{SST} \\ \psi_{\mathrm{ocn}}\end{array}\right)$$

where A and B are large, sparse matrices. Thus, finding the eigenmodes of the differential equations is equivalent to solving a large, sparse, generalized eigenvalue problem. In particular, we care only about the eigenvalues (i.e., the growth rates) with positive real part. This problem can be readily solved numerically, using the Arnoldi method. If positive real eigenvectors exist, the mechanism discussed in Goodman and Marshall (1998) can produce decadal variability in this much more complicated system.

I began working on this problem last fall, and made considerable inroads in gathering background and setting up the problem. I am fairly confident I can solve the remaining problems and obtain results in a couple of months.

   
3.2 Maximal Excitation of Variability Patterns

By the time this project is finished, the adjoint to the MIT parallel coupled GCM should be operational. Using ``adjoint thinking'', we can turn the sort of atmospheric response study done by Palmer and Sun, Kushnir and Held, and Latif et al.on its head. Rather than supplying an SST anomaly and asking if the resulting atmospheric response resembles the NAO or PNA in some qualitative sense, we can form a cost function by projecting the model state onto a pattern of atmospheric variability (i.e., the NAO or PNA) and ask ``upon what forcing variables is this cost function sensitive?'' That is, we can compute the diabatic heating field or SST pattern which maximally excites these modes of variability. If this pattern projects substantially onto the pattern of anomalous SST forcing, it suggests that these modes are indeed reacting to surface forcing, providing evidence for feedback from ocean to atmosphere. A similar study can be done with the ocean model to find out what thermal/mechanical forcing pattern maximally excites observed patterns of SST variability; are the atmospheric and oceanic sensitivity patterns compatible with one another?

3.3 Coupled Growing Modes in a full GCM

While these two projects (numerical calculation of growing modes in a simple coupled system and the calculation of maximal excitation patterns) are rather different, they both lead directly toward a third, more difficult project. Do coupled growing modes exist in a full coupled GCM? If so, we have a very strong implication of coupled instability in generating coupled decadal climate variability. The problem is a logical extension of the study in section 3.1, in that it looks for growing modes in a system with complicated dynamics and geometry, and relies on eigensystem methods for its solution. It follows from the study in section 3.2 in that it asks whether the GCM's atmosphere and ocean can each excite the other to produce a growing mode, and it relies on tangent/adjoint models of a coupled GCM for its solution.

To find growing modes in a GCM, we use the TAMC to construct a tangent linear model L to the GCM: this tells us how small perturbations about a basic state (a particular model run) will evolve in time. That is, if we begin with a perturbation $\psi_0$ at t=0, the linear transformation $\psi(t) = \mathbf{L}(t)\psi_0$ gives the perturbation's state at some later time (assuming small perturbations). We can find coupled growing modes in one of two ways.

The first approach is to look for perturbations which grow while preserving their shape: $\psi(t) = \lambda \psi_0 = \mathbf{L} \psi_0$. This means finding the eigenvectors of the ``matrix'' L. Explicitly writing down L in matrix form and solving for its eigensystem is probably impossible, since the matrix has as many rows and columns as there are variables in the model state. However, the Arnoldi method for finding eigenvalues of sparse matrices does not require the matrix explicitly: it only requires an algorithm for operating L on a vector -- i.e., running the tangent linear model. The Arnoldi method is very efficient, requiring only ``a few'' matrix multiplications to converge on the desired eigenvectors. However, since each matrix multiplication is a run of the tangent linear model, this experiment might be computationally very expensive.

An alternative method for finding growing modes is to compute the singular vectors of L: this gives the fastest-growing modes without requiring preservation of shape. The singular vectors are also often easier to compute, though we must also find L^*, the GCM's adjoint.

Several techniques, such as computing Green's functions for L, may reduce the order and increase the tractability of the computation. Also, there are potential difficulties caused by ``pollution'' of the decadal growing modes of interest by more rapid fluctuations. Solutions to these problems may only become clear after further research; the first two problems may provide some insight.

This problem is quite difficult, and work on it may not be completed within the limited time-frame of a thesis project. Fortunately, both of the two preceding projects are tractable, relevant, and finite in scope. A publishable paper will probably result from each within two years' time, providing sufficient thesis work even if the final project is not completed.

4. Summary

Figure 1 summarizes the relationship between the experiments described in this proposal. MIT's experience with coupled and adjoint modeling provides the background for the maximal excitation / adjoint problem, while my completed work with Dr. Marshall directly leads to the search for growing eigenmodes in a system with complicated geometry. Both of these projects help to provide the knowledge base and technical experience necessary to look for coupled unstable modes in a full GCM. My work last summer towards solving the complex-geometry eigenmodes problem suggests that it should take just a few more months to obtain solutions: interpretation of the results and preparation of a paper should take 2-3 more months. I have less experience with adjoint techniques, so the maximal excitation problem could take slightly longer. This leaves all or most of the '99/'00 school year to work on the GCM coupled modes problem (or to correct a massive underestimate of the time required to complete the first two problems).

 

Figure 1: Relationship of proposed experiments

 

Figure 1: Relationship of proposed experiments

Bibliography

1

D. Cayan.
(a) latent and sensible heat flux anomalies over the northern oceans: the connection to monthly atmospheric circulation.
Journal of Climate, 5:354-369, 1992.

2

C. Deser and M. Blackmon.
Surface climate variations over the North atlantic ocean during winter: 1900-1989.
Journal of Climate, 6:1743-1753, 1993.

3

J. C. Goodman and J. Marshall.
A model of decadal middle-latitude atmosphere-ocean coupled modes.
To appear in Journal of Climate, 1998.

4

Y. Kushnir.
Interdecadal variations in North Atlantic sea surface temperature and associated atmospheric conditions.
Journal of Climate, 7:141-157, 1994.

5

Y. Kushnir and I. M. Held.
Equilibrium atmospheric response to North Atlantic SST anomalies.
Journal of Climate, 9:1208-1220, 1996.

6

M. Latif et al.
A mechanism for decadal climate variaility.
In Decadal Climate Variability: Dynamics & Variability, volume 44 of NATO ASI series, Series I: Global Environmental Change, pages 263-292. Springer, 1996.

7

J. Marshall and F. Molteni.
Toward a dynamical understanding of planetary-scale flow regimes.
Journal of the Atmospheric Sciences, 50(12):1792-1818, 1993.

8

B. Qiu and F.-F. Jin.
Antarctic Circumpolar Waves: an indication of ocean-atmosphere coupling in the extratropics.
Geophysical Research Letters, 24(21):2585-2588, 1997.

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Doctoral Thesis Proposal:
Looking for Coupled Midlatitude Climate Variability using Adjoint and Eigensystem Techniques

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