NCAR/TN-344+IA NCAR TECHNICAL NOTE II

November 1989

Theory and Application of Nonlinear Normal Mode Initialization

RONALD M. ERRICO

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CLIMATE AND GLOBAL DYNAMICS DIVISION

I II

NATIONAL CENTER FOR ATMOSPHERIC RESEARCH BOULDER. COLORADO

TABLE OF CONTENTS List of Tables . . . . . . . . . . . . . . . . . . . . . . . . .. . aiv . . . List of Figures . . . . . . . ... . . . . . . . . . . . . .. . . . . Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . ....... Acknowledgments ...................... . . l. . ...

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vi vii

Part I. INTRODUCTION 1. Reasons for Initialization and Normal Mode Analysis 2. Brief History of Initialization . . . . . . . . . .. 2.1 Initialization before development of NNMI . . 2.2 Development of NNMI ............ 2.3 Problems and impacts of NNMI . . . . . . . 2.4 NNMI as a tool for understanding . . . . . ..

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Part II. DERIVATION OF NORMAL MODES 3. Presentation and Linearization of Model . . . . . . . . . . . . . . . . . 3.1 Nonlinear primitive equations . . . . . . . . . . . . . . . . . .. 3.2 Linearization of equations . . . . . . . . . . . . . . . . . . .. 4. Solutions of Linearized Equations . . .... ..... ..... ... 4.1 Vertical structures ...... . . . . . . . . . . . . . . . . .. 4.2 Horizontal structures ..... . . . . . . . . . . . . . . . . . . 4.3 Field interactions . . . . . .. ................... 4.4 Sequence of transformations . . . . . . . . . . . . . . . . . . . 5. Dynamics of Linearized Model . . . . . . . . . . . . . . . . . . .. 5.1 Linear geostrophic adjustment . . . . . . . . . . . . . . . . .. 5.2 Linear initialization . . . . .. . . . . . . . . . . . . . . . . . .

. 14 . 16 . 17 21 . 22 . 26 28 . 32 . 35 . 36 . 39

Part III. NONLINEAR CONSIDERATIONS 6. 7. 8. 9. 10. 11.

Nonlinear Normal-Mode Equations ............ Scaling Arguments ................... Dynamics of Nonlinear Model .............. Machenhauer's Normal-Mode Balance Scheme ....... Physical Interpretation of Machenhauer's Balance Condition Determination of p ................ ...

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42 45 48 51 52 55

Part IV. APPLICATION OF NNMI TO NUMERICAL MODELS 12. 13. 14. 15.

. . . .61 ........... Vertical Modes for Vertically Discrete Models . a69 Explicit NNM I . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . .. . 73 . . . . . . . . .. Implicit NNMI . . . . . . . .. . . . .77 Further Considerations ..................... . .. . 77 15.1 NNMI vs. no NNMI .................... . . . 79 15.2 Selection of modes to initialize ................ . . . .82 ................ 15.3 Diabatic vs. adiabatic NNMI ... . 83 ............ 15.4 Choice of starting iterates ...... 83 . ... 15.5 Consequences of incorrect mode determination ......... Part V. NNMI AND QUASI-GEOSTROPHIC THEORY

. . . . 16. Scale Analysis in Terms of a Rossby Number . . . . . 17. Scale Analysis of Nonlinear, Diabatic Model Simulations . .. 17.1 Global model results ............... . . 17.2 Mesoscale model results .............. . . 18. Description of Gravitational Modes as Oscillators . . . . . . ....... 18.1 Demonstration of short-term behavior . . . . 18.2 Demonstration of long-term behavior . .. . . 19. Mode Forcing, Interaction, and the Slow Manifold . . ... 19.1 Stability of geostrophic waves and effects of dissipation · . . .. . .... .... . . . . .. 19.2 Slow Manifold . . ...

86 . . . . ... . 89 . . . .... . . . . . .. 91

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109 110

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122 126

Part VI. CONCLUSION 20. Summary

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Appendix A: List of Mathematical Symbols ......... Appendix B: Determination of Nonlinear Interaction Coefficients References . . . . . . . . . . . . . . . . . . . . . . . .

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LIST OF TABLES Table 12.1. Values of a for data levels, standard atmospheric values of T at those levels, and the set of equivalent depths H, determined for the standard CCM1. Note that t is a a-level index for a and T, but refers to the ordering of the vertical modes in the case of H.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

Table 12.2. Values of a for data levels, standard atmospheric values of T at those levels, and the set of equivalent depths H, determined for the standard 10-level MM4. Note that

e is a a-level of H.

index for a and T, but refers to the ordering of the vertical modes in the case

. . ..

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

Table 12.3. The matrix 0 which indicates the nonorthogonality of the vertical modes in the MM4 for a standard atmosphere and 10 equally-spaced u-levels. The index t is listed both vertcally and horizontally. An element of 0 with horizontal index i, and vertical index t2 describes the projection of mode t4 on mode t2, and vice versa

.....

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LIST OF FIGURES Fig. 1.1 The time series of surface pressure as forecast by the CCM1 for a point near Eureka, California . . . . . . . . . .. . . . . . . . . . . 4 Fig. 12.1 The structures of the vertical modes of the standard 12-level CCM1 determined for a standard T profile ........................ 65 Fig. 12.2 The structures of the 10-level MM4 with equally-spaced a-levels for a standard T profile . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .66 Fig. 13.1 Time series of p,,T, a, g at a CCM1 grid point near Eureka, California as forecast starting from a noninitialized and initialized analysis .. . . . . . 72 Fig. 14.1 Time series of p,,u,T, and w at a point near Detroit, Michigan for 24-hour forecasts begun from an initialized and noninitialized analysis ........ 76 Fig. 17.1 Spectra of normalized rms magnitudes of terms in the mode coefficient tendency equation for selected sets of modes ................... 94 Fig. 17.2 Spectra of normalized magnitudes of terms in the 6-tendency equation at hour 48 of a simulation for selected vertical modes .............. . 97 Fig. 17.3 Spectra of normalized magnitudes of terms in the f - Vtendency equation at hour 48 of a simulation for selected vertical modes . . . . . . . . . . 98 iv

Fig. 18.1 Harmonic dials for selected modes in a noninitialized CCM forecast beginning from an ECMWF FGGE analysis ................... 101 Fig. 18.2 Response function R for the forcing of eastward propagation and westward propagation for a wave with resonance at a 33 hour period (eastward direction) and linear e-folding damping period of 5 days . . . .............. . 104 Fig. 18.3 Harmonic dials of four modes obtained near the end of a long climate simulation . .. . .. . .. . . . . . .. . . . . . . . . . . . ... 106 Fig. 18.4 Power spectra of modes corresponding to Fig. 18.3

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Fig. 19.1 A schematic representation of the slow manifold in a simple two-component model . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 115

v

PREFACE

Just prior to being appointed an Adjunct Associate Professor in the Department of Meteorology at the University of Utah, I was asked to present a series of lectures on the subject of normal mode initialization. These were delivered in May 1987 during a two-week period. Approximately 20 students and faculty attended the 6 one-hour lectures. More condensed versions of these lectures have been presented at other institutions, and still others have asked for copies of my notes. This technical note is derived from these notes and from several of my papers on the subject of normal mode initialization, but its form is more readable than the notes and more succinct than the collection of papers. The above situation provided the possibility of preparing this technical note, but the motivation was not simply to publish a set of notes. During my fifteen years of work on the subject I have come to realize that normal mode initialization is not just some "trick" used to remove forecast noise. Rather, the subject is basic to dynamic meteorology. In fact, most principles, limitations, and extensions of quasi-geostrophic theory can be derived from this theory in a straightforward and elegant manner. Also, since it uses a linear theory as a basis to investigate nonlinear behavior, it is simple to comprehend but not restricted to linear or quasi-linear contexts. For many studies, the normal modes themselves or their associated initialization theory in general provide a valuable tool for investigating and understanding complex atmospheric behavior. For these reasons, it is my firm opinion that the theory and principles of normal mode initialization should be part of the core curriculum of graduatelevel dynamic meteorology. Although not as detailed as a textbook, this technical note is intended as an aid in such a curriculum.

Ronald M. Errico NCAR September 1989

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ACKNOWLEDGMENTS

Although I was not well aware of all the connections at the time, the subject of this technical note was also the subject of my Ph. D. thesis. The question investigated in that thesis was "Why is the atmosphere nearly quasi-geostrophic", for which I thank my thesis advisor E. N. Lorenz for posing to me. Since the thesis was completed in 1979, it has been difficult for me to stray far from the topic because of its many applications and intriguing aspects. While at NCAR I have benefited greatly from the presence of many colleagues who have contributed greatly to the theory and application of normal mode initialization. Among them were (at various times) the members of what is now called the Global Dynamics Section of the Climate and Global Dynamics Division at NCAR. These were R. Daley, D. Williamson, A. Kasahara, J. Tribbia, and C. Leith. The PSU/NCAR mesoscale model was provided by courtesy of R. Anthes and colleagues. The CCM was provided by D. Williamson and colleagues. The Navy models and analysis were provided by E. Barker, R. Gelaro, and colleagues. The lectures at the University of Utah were at the invitation of J. Geisler, and J. and J. Paegle. The more abbreviated lectures at the Naval Postgraduate School were at the invitation of R. T. Williams. F. Carr at the University of Oklahoma and R. Daley also encouraged me to distribute a readable version of my notes. R. Bailey provided invaluable assistance in preparing the manuscript. G. Bates provided assistance in preparing final versions of the figures. Both he and B. Eaton also aided me at various times in preparing normal mode software for use at NCAR.

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Part I. INTRODUCTION According to its title, this report is intended to serve as a comprehensive summary of the theory and application of nonlinear normal mode initialization (NNMI). NNMI was developed in the late 1970s for initialization of models used for numerical weather prediction (NWP). It was originally developed and described in terms of normal modes of linearized versions of the models; i.e., in terms of the independent solutions to particular eigenvalue problems. In contrast to earlier initialization schemes using normal modes, terms previously treated as nonlinear, and therefore neglected in the eigenvalue problems, were reconsidered by NNMI. Although many questions and details regarding its application remained unanswered, NNMI proved quite successful as an initialization procedure, and during the 1980s it or its derivatives became the standard initialization technique. In the future, as observation, analysis, and forecast systems improve, it is not certain that NNMI will remain an appropriate initialization procedure. However NNMI should not be regarded as only an engineering tool which may or may not be useful in some NWP system.

Rather, the theory and framework of NNMI is fundamental to the dynamics

of forecast models and the relationships between various types of atmospheric data. In fact, the theory can be considered as more fundamental than quasi-geostrophic theory, since the latter may be considered as a low-order approximation to NNMI theory, and since NNMI theory reveals the processes which maintain quasi-geostrophy in both models and the atmosphere. Also, since it is based on linear and quasi-linear concepts, NNMI is relatively easy to interpret, and its framework provides a useful tool for the analysis of model responses to both data input and internal forcing. Almost all global atmospheric data analyses and many regional data analyses are produced by data assimilation systems. These systems use a numerical forecast model to interpolate or extrapolate information into data-void regions through production of shortterm forecasts. Most systems also use some form of NNMI. For this reason, the possible

1

effects of NNMI should be considered when interpreting these analyses, even in a non-NWP context. In some cases, the effects of NNMI on even large-scale time-mean fields may be quite profound (c.g., see Rosen and Salstein, 1985). The relevance of understanding the principles of NNMI, therefore, currently extends well beyond the subject of NWP. In this report, the word "initialization" is used exclusively in the restricted sense of an adjusting or constraining of data for use as initial conditions in model forecasts, especially for NWP. The process of producing data on some regular grid or in terms of coefficients of structure functions (e.g., spherical harmonics) using irregularly-spaced observations is called "analysis.

Such analysis is necessary to begin a numerical forecast, since models

necessarily represent data in some structured way. (However, the word "analysis" will not be used exclusively in this sense.) An analysis must always be performed prior to producing a forecast, but an initialization is not strictly necessary. Discussion of the analysis problem is not an intention of this report, except as it relates to some aspects of the initialization problem. For discussion of the analysis problem see, e.g., Bengtsson et al. (1981). In Part I of this report, the purpose of initialization is discussed, followed by a brief history of solutions to the problem of initialization. Attention is focused on NNMI rather than on earlier schemes. In subsequent parts: (Part II) a simple model is used to introduce and describe normal modes and the phenomenon of linear geostrophic adjustment; (Part III) nonlinear aspects are reintroduced and NNMI described; (Part IV) the applications of NNMI to global and regional models are presented; (Part V) and the relationship of NNMI to quasi-geostrophic theory is discussed. A summary stressing the use of NNMI is presented in Part VI.

1. Reasons for Initialization and Normal Mode Analysis The motivation for using NNMI is discussed in this section. Since this section is intended as an introduction only, most literature citations will be omitted. Instead, other sections of this report will be referenced, and details and citations will be found in them. 2

An example of a time series of surface pressure near Eureka, California as forecast by an NWP model is presented in Fig. 1.1. The model is version 1 of the NCAR Community Climate Model (CCM1), initialized with an analysis produced for OZ 16 January 1979. The forecast is described more completely in Section 13. The time series produced when no initialization has been performed is indicated by the solid line in Fig. 1.1. Note that the behavior of the surface pressure has two primary characteristics: there is a gradual increase of approximately 10 mb during 24 hours, and there is a superimposed, rapid oscillation with changes as large as 5 mb in two hours. Similar characteristics of the surface pressure forecast are observed at all other locations in the model during this forecast period. Also, most other dynamic fields are similarly characterized as having both slow and fast components, but to greater or lesser degrees (for some examples, see Section 13). A barograph of the verifying observed surface pressure for the same time and place as Fig. 1.1 has not been prepared. Generally however, fast-changing components of the surface pressure with amplitudes as large as in Fig. 1.1 are rarely observed anywhere, whereas in the CCM1, they are observed during the first day (or longer) for nearly all forecasts, everywhere. Clearly, the forecast of these components is unrealistic. There are errors somewhere in the CCM1 or in the analysis used to start the forecast. The existence of unrealistically large, high-frequency components in the forecast may or may not have a significant impact on the use made of the forecast (Section 15). Their presence may simply render interpretation of forecast synoptic maps more difficult, or, much worse, they may trigger unrealistic convection and thereby destroy the utility of a forecast. One significant problem has been the degrading of analyses which use noisy, short term model forecasts as a source of data (background information) in addition to observations. Clearly, even if this degradation has little impact on a specific application, it is worthwhile to investigate, understand, and correct the problem in order to properly

3

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FIG. 1.1 The time series of surface pressure as forecast by the CCM1 for a point near Eureka, California. The solid line is from a forecast with no NNMI; the dashed line is from a forecast starting from the same initial analysis, but with NNMI [from Fig. 8.4 in Errico and Eaton, 1987].

assess its impact and to prepare for future systems where smaller errors may increase the relative significance of this kind of initial noise. The amplitudes of the high-frequency components (often called "noise") in forecasts depend on many characteristics of the model and its initial analysis. For this reason, the amplitude shown in Fig. 1.1, although typical of CCM1 forecasts, should not be considered typical of other models. However, unless the noise problem is specifically rectified, all primitive-equation forecasts have such noise to some noticeable degree. The amplitude of noise in the surface pressure field generally decreases with time, although the same may not be true for some other fields. In global models, the rate at which noise decreases will depend on characteristics of the model's physical and numerical dissipation (Errico and Williamson, 1988), and in limited-area models it also will depend on the domain size and the formulation of lateral boundary conditions (Warner et al. 1984; Errico and Baumhefner, 1987). Some of these points are discussed in Sections 17 and 19. Actually, the source of this noise has been known for a long time (Section 2). Most NWP models admit inertial-gravitational waves as components of their general solutions. Analysis inaccuracies will usually result in the presence of such waves in subsequent forecasts, unless some adjustments of the data are made. The process of adjusting the data for this purpose is called initialization. It is related to the process of geostrophic adjustment, as discussed in Sections 5 and 8. Many different but related initialization schemes have been developed since the advent of NWP, but the most successful one has been NNMI. Before proceeding to discuss initialization, it is pertinent to mention that other solutions to this unrealistic wave problem have been used. Quasi-geostrophic models do not admit such waves in their forecasts, although they also miss other effects due to their filtering of even slow inertial-gravitational components. Other filters, such as selective diffusion have been applied to NWP models, for which a difficulty is to restrict their effects to only the unrealistic components.

Filters have also been applied successfully

5

to forecasts themselves; i.e., after forecasts have been produced by the models (e.g., Williamson and Temperton, 1981; Kuo and Anthes, 1984). Of course, if any unrealistic waves have impacted the slower components of the forecast, their impact is not removed by such subsequent filtering. The method of NNMI is described primarily in Parts II-IV and its application specifically to the forecast in Fig. 1.1 is described in Section 13. However, for the purpose of putting what follows in perspective, the results of applying NNMI to that forecast are also presented Fig. 1.1 as the dashed line. Note that the initial surface pressure at this location has been reduced by approximately 1 mb. Thereafter, the time series from the initialized forecast is almost precisely that which would be created by subjectively smoothing the noninitialized time series: the slow pressure rise with time appears to be reproduced almost precisely as seen in the noninitialized forecast, but any superimposed high-frequency oscillations are almost undetectable.

It should be emphasized that the

differences between the two time series have been produced by starting the forecasts from only slightly different initial conditions and not by any subsequent alteration of the model. The NNMI technique explicitly considers the structures and behaviors of inertialgravitational waves. Therein lies one cause of its successful application. However, this consideration also renders aspects of the NNMI technique useful for many analyses of model behavior, especially when slow and fast model components are to be formally distinguished. For this reason, the theory and application of NNMI should remain useful even beyond a time, if ever, when initialition is no longer explicitly useful for NWP.

2. Brief History of Initialization The intention of this section is to provide a brief overview of initialization in general and NNMI in particular, as well as examples of the use of normal modes as a tool for understanding model results. A good review of initialization prior to NNMI may be found in Bengtsson (1975) while NNMI has been reviewed by Daley (1981a).

6

The use of the

NNMI framework as an analysis or theoretical tool has not be reviewed in any detail. In this section, many contributors to the development of initialization and NNMI will likely be neglected. Further citations may be found throughout this report as well as in the lists of references in the cited literature.

2.1 Initialization before development of NNMI The need for initialization was first observed in Richardson's (1922) famous experiment where he attempted to use the primitive equations for producing a numerical forecast of weather in Europe. He lacked sufficient data (there was no rawindsonde network at that time), with the result that forces determined from his initial fields were so in error that huge dynamic tendencies were obtained. These tendencies were so unrealistic that Richardson abandoned his attempt. It was not until the advent of the modern computer, the development of quasi-geostrophic theory by Charney (1948) and others, and a global observation network, that numerical forecasting in general, and the use of the primitive equations in particular, were re-attempted. For discussion of some aspects of this history, see e.g., Platzman (1987). The presence of Richardson's large initial tendencies is related to the phenomena of geostrophic adjustment (Section 5). The principles of linear geostrophic adjustment date back at least to Rossby (1937, 1938) who showed how to derive a final geostrophically balanced state from initial conditions in a barotropic atmosphere (see also the review by Blumen, 1972).

Hinkelmann (1951) and Charney (1955) related the presence of large

initial tendencies in numerical forecasts to this adjustment process: Essentially, a portion of the errors in the initial conditions are interpreted by the model as due to the presence of (unrealistic) inertial-gravitational waves which subsequently propagate throughout the forecast and appear as meteorological (gravitational) "noise". Larger analysis errors tend to yield greater noise. Charney (1948) earlier had shown how to derive a set of equations which excluded such noise (the quasi-geostrophic equations), however it was clear that some 7

important meteorological activity was thereby modelled poorly (e.g., fronts and tropical circulations). Other exclusionary equations were also developed (e.g., the semi-geostrophic equations; Eliassen, 1948; see also Hoskins, 1975), but some appropriate initialization scheme for using the primitive equations without unrealistic noise remained desireable. Charney (1955) showed that gravitational noise could be reduced within forecasts which used the primitive equations by constraining the initial condition to satisfy a nonlinear balance equation. Satisfaction of a linear geostrophic relationship also may reduce noise, but the nonlinear balance equation produced significantly better results. Charney's balance equation concerned only the rotational part of the wind, however Phillips (1960) showed that consistency required that the divergent part of the wind should also be constrained to satisfy a kind of balance equation, specifically a form of the quasi-geostrophic omega-equation. These results of Charney and Phillips are discussed in terms of NNMI in Section 10. Miyakoda and Moyer (1968) and Nitta and Hovermale (1969) developed methods which effectively filtered gravitational noise from primitive-equation forecasts by using the model dynamics and numerical scheme. Their methods have been denoted as dynamic initialization schemes, since they require time integration of the equations in order to specify even the initial condition. These methods should be contrasted with the previous static schemes which only applied balance condition constraints at the initial time. The dynamic schemes were sufficiently successful so that primitive equation models became the standard for numerical weather prediction. However, because the filters used by those methods were not so selective as to affect only components responsible for the noise, these early schemes had drawbacks, notably including a general weakening of the entire circulation. Recently these dynamic schemes have been recast in the framework of NNMI (Sugi, 1986), with many of the earlier drawbacks diminished.

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2.2

Development of NNMI In many contexts, waves are quasi-linear phenomena, and it had been known for a

long time that one class of solutions to the linearized primitive equations described inertialgravitational waves. Dickinson and Williamson (1972) used that knowledge to define an initialization scheme which attempted to filter high-frequency waves from model initial conditions. This method has since been called linear normal mode initialization. It was not very successful. The reason for this result is discussed in Section 8, and primarily regards the neglect of consideration of nonlinear effects on the gravitational modes. Nonlinear balance equations were first described in terms of normal modes by Machenhauer (1977) and Baer (1977).

Machenhauer considered the prognostic equations for

amplitudes of gravitational modes, and showed that with or without initialization, the adiabatic nonlinear forcing term has a strong, slowly varying component. This yields a correspondingly slow response, which approximately satisfies a nonlinear balance equation expressed in terms of the normal mode amplitudes and their forcings.

He proceeded

to show how solutions to this new balance condition could be determined and applied to the initialization problem.

Baer applied a Rossby number scaling to the primitive

equations schematically expressed in terms of the normal modes and explicitly considered the presence of multiple time scales. He showed that asymptotically slow solutions were possible, and that these solutions were characterized by a nonlinear balance condition expressed in terms of the normal modes.

This work was later extended by Baer and

Tribbia (1977) to the practical application of this result to the initialization problem. The methods of Machenhauer, Baer, and those who built on their work, are collectively called nonlinear normal mode initialization schemes. They were applied to some global forecasting systems by Andersen (1977), Daley (1979), and Williamson and Temperton (1981), among others, and also to regional models by Briere (1982), Du Vachat (1986), and others. Machenhauer's scheme was used predominately, although the scheme of Baer and Tribbia provided a more suitable theoretical framework for many problems.

9

Soon after NNMI was presented by Machenhauer and by Baer, it became apparent that the NNMI balance equations were related to the nonlinear balance equations of Charney (1955) and Phillips (1960), but it was unclear what that precise relationship was. Leith (1980) used the context of an f-plane model to show their specific relationship. Bourke and McGregor (1983) subsequently developed an NNMI scheme which only explicitly considered the vertical structures of the modes, resulting in a simpler application of NNMI to regional models for which horizontal mode structures were more difficult to determine. Temperton (1988) has presented an elegant mathematical demonstration of the equivalence of the Bourke and McGregor method with that of Machenhauer applied to the same model. He has termed schemes which use only the vertical structures "implicit" NNMI schemes, as contrasted with "explicit" schemes which require determination of the modes' complete three-dimensional structures. Many other initialization methods have been developed since the advent of NNMI. Some are described throughout this section and this report where applicable. For all these schemes, NNMI provides a benchmark with which to compare results and a theoretical framework with which to explain methodologies and reasons for success.

2.S Problems and impacts of NNMI There are several works which discuss various aspects of the results of NNMI, regarding its effects on both analysis systems and forecasts. Daley (1979) showed that precipitation forecasts were not greatly improved in his model by the incorporation of NNMI, which was a reminder that NNMI could only definitely improve those components of the forecast which it was specifically designed to affect (i.e., gravitational noise). Bengtsson (1981) revealed that some applications of adiabatic NNMI tended to weaken tropical circulations. Wergen (1983) introduced a diabatic NNMI scheme to alleviate that tendency. Essentially both he and Bengtsson reasoned that since tropical circulations are diabatically driven, adiabatic NNMI would fail to produce realistic tropical circulations. Errico (1984b, 1989a, 1989b), Errico and Rasch (1988), and Errico et al. (1988) suggested that the poor results of 10

applying adiabatic NNMI in the tropics were not only due to neglect of diabatic processes there, but equally to the inappropriateness of the NNMI balance condition itself within the tropics (at some scales; see Part V). Another problem noted in the applications of NNMI was the lack of general convergence of Machenhauer's (1977) scheme for obtaining iterative solutions (e.g., Williamson and Temperton, 1981). The problem was discussed using a one-dimensional model by Ballish (1981) and in more general contexts by Errico (1983) and Rasch (1985a). Rasch (1985b) and Lynch (1985) introduced alternative schemes with better properties for obtaining the desired NNMI fields. Kitade (1983) introduced an under-relaxed version of Machenhauer's scheme, but its improved convergence was limited for reasons discussed by Errico (1983). Thaning (1983) showed that in some cases, multiple or inappropriate solutions may exist, although his context was for a simple model and high Rossby number. The question of which horizontal and vertical scales should be modified by NNMI also has received much, although insufficient, attention. For most NNMI schemes, the set of initialized modes or scales is restricted in practice by a lack of general convergence of their iterative solutions as previously discussed. Puri and Bourke (1982) and Puri (1983, 1985, 1987) examined relationships between convection and initialization. Errico (1984b, 1989b, 1989c), Errico and Rasch (1988), and Errico et al. (1988) used model simulations in an attempt to deduce which scales are balanced in the atmosphere. Carr et al. (1989) compared forecasts produced with different scales initialized, and showed that a more restricted set of modes performed better with their particular NNMI scheme and forecast and analysis system. Initialization is most important in the context of data assimilation, as reviewed by Bengtsson (1975, 1981) and discussed in Section 15.

Static balance constraints were

considered by Sasaki (1956) in the context of variational analysis methods.

Flattery

(1967) considered analysis of the rotational normal modes. Daley (1980), Williamson et al. (1981), Tribbia (1982), and Williamson and Daley (1983) discussed the appropriateness

11

and methodologies of incorporation of NNMI into analysis schemes. This was later cast in a simpler framework by Temperton (1988) and Fillion and Temperton (1989). Lorenc (1986) discussed optimal use of nonlinear relationships of the data in the context of Bayesian analysis.

£.4 NNMI as a tool for understanding The concepts developed for NNMI have also proved useful for investigating basic questions regarding atmospheric behavior. One such question concerns the reasons for quasi-geostrophy in the atmosphere. A succinct answer to this question is presented by Charney (1955) in his introduction (and discussed in Part V). However, he provided no direct support for his conjectures other than the observation of quasi-geostrophy and the general knowledge of atmospheric processes. Errico (1979, 1981, 1982a, 1984b, 1989a), examined the aspects of stability, dissipation, and time-scale interaction in Charney's argument and showed them to be correct. A more mathematically complicated but elegant approach was taken by Lorenz (1980) and Leith (1980) who developed the concept of the slow manifold. Essentially, their conjectures were that, although fast, quasi-linear gravitational waves are solutions to the primitive (or similar) equations, the nonlinear interactions and strong external forcing of slow components result in solutions in which these fast waves are absent.

The accuracy of their conjectures remains questionable

(Krishnamurthy, 1985; Warn and Menard, 1986; Lorenz, 1986; Lorenz and Krishnamurthy, 1987; Errico, 1982a, 1989a), however the concept has proved useful as an approximation (Daley, 1981; Tribbia, 1982). Further discussion of this topic appears in Section 19. The use of NNMI concepts for investigating model dynamics has already been discussed in Section 2.3. This use has not been limited to those studies mentioned. Daley and Puri (1980) and Kalnay et at. (1986) showed that lack of consideration of geostrophic adjustment and NNMI concepts can lead to limited usefulness of temperature data (c.g., as derived from satellite observations) when wind data is not also used. Daley et al. (1981) investigated forecast errors produced by artificial walls placed at some latitudes in 12

models. The importance of NNMI for understanding predictability studies was emphasized by Errico and Baumhefner (1987). Model normal modes have also been used for climate studies. Paegle et al. (1986) used them to investigate interactions between the tropics and extratropics in studies of the atmospheric response to sea surface temperature anomalies. Gelaro (1989) used them in a different manner for a similar investigation. Kasahara and Puri (1981) and Tanaka and Kung (1988) have examined normal mode coefficients determined from atmospheric analyses. Branstator (1989) has demonstrated that consideration of normal modes of complicated basic states can yield substantial insight into low frequency behavior and seasonal climate. These newer works suggest that analyses using normal modes will continue to be useful for understanding of atmospheric behavior on many time scales.

13

Part II. DERIVATION OF NORMAL MODES

In this part, the normal mode equations are presented.

They are derived for a

model defined on a plane with periodic boundary conditions, although the differences and similarities with modes on a sphere are discussed at the end of Section 4. The application to a periodic domain on a plane greatly simplifies the transformation between normal-mode and physical-space descriptions, and therefore is very appropriate for pedagogic purposes (Leith, 1980). The complete adiabatic model is presented in Section 3 along with an appropriately linearized version. The solutions of the linearized version are then determined from an eigenvalue problem in Section 4. Their properties are discussed in Section 5 in the contexts of the process of linear geostrophic adjustment and the technique of linear initialization. The usual notation is used where no conflicts occur, and any unusual or ambiguous notation is defined when introduced. For reference, a variable list appears in Appendix A.

3. Presentation and Linearization of Model The model uses the primitive equations. These equations permit the propagation of gravity waves, but not sound waves. Equations which filter gravity waves, such as the quasi-geostrophic equations, would not yield the types of modes in which we are interested. The horizontal coordinates are a Cartesian z (increasing eastward) and y (increasing northward) with corresponding velocity components u and v. The domain is specified as periodic in both directions, with some fundamental wavelength LF; i.e., for any field a,

14

a(x,y) = a(x, y + LF) (3.1) =a(x + LF, y).

This condition is not strictly necessary, but allows the use of simple trigonometric functions for the description of the normal modes. More general boundary conditions for limited-area models are discussed in Briere (1982). However, (3.1) corresponds closely to the condition on the sphere (which is also a periodic domain). Condition 3.1 also places restrictions on the z, y dependence of the Coriolis parameter f if solutions are to remain periodic for all time. However, since many types of conditions and approximations are possible, none is explicitly stated here, since such details are unnecessary for all the derivations to be made below. Most simply, we may just consider that f = fo everywhere, but at the end of Section 4 we will describe important effects due to variable f on the sphere. The vertical coordinate is sigma, defined as a = -, P.

(3.2)

where p is the pressure at some height and p, is the pressure at the surface. The top and bottom of the model are therefore a = 0 and o = 1, respectively. The use of other vertical coordinates may change details of the form of the equations, but all qualitative comments regarding the form of the solutions must apply equally well to other coordinates; i.e., the properties of the solutions are not altered by a coordinate change, although the exact expression of those properties may be altered. The direction perpendicular to a surface of constant a will be simply called the vertical, and that surface will be called horizontal. Terms which describe the diabatic processes are simply denoted by D with subscripts to denote the equations to which they are applied. Different diabatic processes will not be distinguished until Part V.

15

S.1 Nonlinear primitive equations The nonlinear equations are

au

au

at av

at aT

au

+

UT

+a 6-fu-

+ v

aT

a In(p,/p) at

-

j-

+fv--

o

aT

u-+ x

at

.9u

uxV + V y

aT

V-+ V a-+ ady aa

a

[O + RT ln(p./p)] + Dv

(3.3)

[

(3.4)

+ RTln(p./p)I+Dv

T-+ +DT

p

I 6da

(3.6)

Jo

RT

, _|r 6 - aI

0=

p

ay

-

v=a

(3.7)

d

(3.8)

a

ao

E da -

, .u

v

(x

9y

(3.5)

6 da

(3.9) (3.10) (3.11)

The first four equations are prognostic and the remaining ones diagnostic. These equations are derived from the conservation of momentum (3.3-4), the first law of thermodynamics (3.5), the conservation of mass (3.6), and the hydrostatic relationship (3.7).

The

variable a may be interpreted as a counterpart to the vertical component of velocity in acoordinates; 6 is the velocity divergence on a a-surface, hereafter simply called divergence; and

¢

is the vertical component of relative vorticity, hereafter simply called vorticity. At

this point, the use of 6 and $ may simply be considered as abbreviations for the sums and differences of the indicated differentials.

16

9.2 Linearization of equations The propagation of many waves in the model and atmosphere may be considered as a quasi-linear phenomenon, as demonstrated in Sections 5 and 18. For this reason, since our particular interest is in gravity waves, it is appropriate to consider a linearized form of the model equations. To do this we define an appropriate basic state (denoted by a bar over the variables) and a perturbation from that state, and ignore any terms which involve products of two or more perturbation variables. Further, we select the basic state such that either it itself is a stationary (i.e., time-independent) solution to the nonlinear equations, or assume that some unspecified external forcing renders it stationary. The selection of the basic state must be done carefully. The two primary considerations are that the resulting linearized equations are solvable by some means (or that at least we learn something from them) and that those equations describe significant aspects of the dynamics. The first consideration motivates us to make the basic state as simple as possible, and the second motivates us to make it as complex as possible to minimize the labeling of dynamic terms as nonlinear with their subsequent neglect when only the linearized terms are retained. Since these are conflicting considerations, selection must depend on the exact purpose of the linearization. For the purpose of NNMI, the linearization primarily is used to separate the dynamic fields into portions which describe either gravity waves or quasi-geostrophic fields. Different basic states will likely yield different separations, so that even if the NNMI scheme directly affects only gravity waves, the actual effect on the dynamic fields will differ as the basic state is varied. This will be further discussed in Section 15.5. In NNMI, the terms neglected in the linearization are not just discarded, but they are instead reconsidered (Part III), so that the linearization acts to distinguish terms rather than filter effects. The most typical basic state chosen for NNMI is a resting, horizontally uniform, convectively stable atmosphere. This may be denoted by

17

u=0

v-0

(3.12)

= T((a) In P. = constant

= ¢(To,) e=0o The last is a condition that there are no mountains. The resulting basic state is a stationary solution to the adiabatic, primitive nonlinear equations.

Also, this basic state yields

separable linear equations which are especially easy to solve, as discussed below. The linearization is about a Inp. rather than about p* because p, explicitly appears in the adiabatic equations for this model only in the form of In p,, and this linearization therefore acts to retain more of the explicit pressure effects on the dynamics without making the equations more difficult to solve. The horizontal means of T(a) and In p, determined from some analysis may be used to define the basic state, or a standard atmosphere may be used. Furthermore, we can define p = exp (in p,). The linearization will be about a constant value of

f,

denoted

fo,

since this greatly

simplifies the solutions to the linear equations. On the sphere this simplification is not made (Appendix B). On the plane, this restriction does affect the linear solutions as will be noted, but the nonlinear initialized solutions are less affected.

Any term involving

a deviation of f from fo will be considered a nonlinear term, to be neglected by the linearization, but reconsidered by the NNMI. The resulting adiabatic linear equations are

fov -

y [' + RTln(p,/p)]

(3.13)

= -f^u^-

[(1 +RTIn(p./p)]

(3.14)

t =

18

AT' T t = -a-°, +cKT-

Qa

at

Oln(p/p) =

(3.15)

P

j|6 do

(3.16)

l e- oRT' a

(3.17)

= -- l

6 do

(3.18)

6= a j

do-

w

po

a = oz

+

Ox

- =v

6 do

(3.19)

o

_

(3.20)

Oy -

^a(3.21) U

where superscript primes on T and b indicate departures from the mean state, but no primes are indicated for u, and v, w, 67, 6, or f since their basic state is 0. These should be compared with (3.3-11), respectively.

Note that the presence of mountains (4,) is

considered as a perturbation effect. The quantity whose gradient describes the linearized pressure gradient force in acoordinates is the pseudo-geopotential p defined as cp = X' + RTln(p,/p)

(3.22)

Its time tendency may be determined by combining all the diagnostic equations and thermodynamic equations to yield the integral-differential equation

9Po aOt

R1

f

OT

[(2

\

T

a- a) l6& d" "] da' -RT(1)

6 da'

(3.23)

The operations on 6 which comprise the right-hand side of (3.23) are denoted as (the operator) r, so that the four prognostic linearized equations may be written simply as = at9 at= 19

-foi(3.24) ' fo xz

at

=-ou-(3.25) - ay

aOv

_-

ai = -r8()

ain(p./)

=

(3.26)

_f

(3.27)

It is also useful to define the differential operator which corresponds to r, written as r-l, which satisfies rr--(a) = r - 1 r(a) = a,

(3.28)

where a is any dynamic field. The operator is r-'(a)

-R -

(3.29)

a r aa,

with boundary conditions

aa

(3.30)

{ _0a at a=1

where

r

KT a

89T a

(3.31)

-

is a mean static stability parameter. If a is the field 6 then the boundary conditions are equivalent to

=

0 at a = 1 and 0.

Equations 3.24-27 are the linearized adiabatic primitive equations. It is easy to show that they conserve the sum of a quadratic form of the kinetic and available potential energies per unit mass, defined as E

JOJj +. JO

+u2+v2] .

dxdy da

(3.32)

As successive transformations of the linearized equations are considered in Section 4, the corresponding expressions for E in terms of the transformed fields will be presented in order to complete the description of each transformation.

20

4. Solutions of Linearized Equations Equations 3.24-26 comprise an eigenvalue problem. Schematically, we may write

[u(x,y,at) d | v(xy,ot)

(tx

[u(x, y,o,t) = C v(x,y, a,t) I,

(4.1)

("(,y, , t)]

y(,y,o,t)

where ZC is a linear operator which includes horizontal derivatives and vertical integrals. If expressed in finite difference form, ZC could be written as a matrix operator. However, because of the simple basic state chosen, it is possible to solve (4.1) analytically. We note that it is not necessary to also consider the linearized prognostic equation for In p./p

(3.27)

since it is already implicitly considered in (3.26). However, if the linearized behavior of p. is to be examined, then (3.27) must be considered, as we will do in Section 11. The form of the operator £ and the simplicity of the boundary conditions (3.1) allows us to obtain separable solutions to (4.1); i.e., we can separate the operations of £ by considering £(x,y,a) =

AC£C£y

£c(H,m,n,A)

(4.2)

where the subscripts indicate the coordinates to which the operators apply and the values H,m, n, A are respective eigenvalues of the operators £,C, ,^Cy, each of £ LRL

-1

c

order (6/fo) < c order (/g9H)

< € n

_fo

where LR is the Rossby radius of deformation. The last four conditions are specified in order to limit our considerations to a manageable set; otherwise too many possible scaling relations exist and no useful conclusions can be made without more detailed consideration of the functions N. Our scaling of the nonlinear terms is consistent with their primarily being advective terms. With consideration of (16.5), the corresponding terms in (16.1-3) have scales (nlfo) E

(6/fo) +

(fl/o) (6/fo)

(n/fo) (P/gH)

+

2,

(16.6)

I/gH + c2,

(16.7)

(6/fo) + €({/gH).

(16.8)

For the scales which contain most of the kinetic and available potential energy in the atmosphere, it is appropriate to consider e> D), except at the smallest horizontal scales where parameterized eddy diffusion is important. Similar qualitative results apply at all forecast times beyond hour 36 and for other synoptic situations, including summer simulations. The results of Errico (1989c) highlight the strong dependence of model generated balance on vertical scale. Large vertical scales generally tend to be balanced and small vertical scales unbalanced, with the imbalances defined by nonnegligible magnitudes of observed tendencies. One important implication is that NNMI-type balance conditions appear to be appropriate only for large equivalent depths. Another implication is that appropriateness of either nonlinear balance equations (Charney, 1955) or quasi-geostrophic omega-equations (Phillips, 1960) for initialization or diagnostic description of the atmosphere is similarly restricted to large vertical scales. In particular, the use of a quasi96

DIVERGENCE TENDENCY

s 48 hours

-

10

|

H=7721. m

-1I

10

I 10

-=

z 4

2

-4

0 2

0K

-5

o 10

I 1 -6 1

in '

10 7

20

40

4

HORIZONTAL SCALE

DIVERGENCE TENDENCY

H=

35. m

2

1

(100km)

DIVERGENCE TENDENCY

t, =48 hours

H=0.167 m

t =48 hours

T 10 0

I

0

i

s.° s4 I 10

4

o z

10 40

20

10 7

HORIZONTAL SCALE

4

2 (100km)

1

40

20

10 7

HORIZONTAL SCALE

4

2 (100km)

FIG. 17.2 Spectra of normalized magnitudes of terms in the 6-tendency equation at hour 48 of a simulation for (a) t = 1, (b) e = 4, and (c) t = 10. Terms are labeled as T (tendency), L (linear), A (adiabatic nonlinear), and D (diabatic) [from Fig. 4 in Errico, 1989c).

1

VORTICITY TENDENCY

10

H=7721. m

t, =48 hours

-1

-2

I'

10

,

,

-3

= 10 z

0

4 -4 10

o N

0I,.O 10 z

-5 I I I

-6

10

10

20

40

7

HORIZONTAL SCALE

VORTICITY TENDENCY

H=

35. m

2

4

1

(100km)

VORTICITY TENDENCY

t, =48 hours

t, =48 hours

H=0.167 m

-3

0 t-

z

0

= 4-

10 10

o

fc 2,,.I N

z

0z .2 I I

lO-

O 41

i

z

10 40

20

10 7

HORIZONTAL SCALE

4

2 (100km)

1

40

20

10 7

HORIZONTAL SCALE

4

2 (100km)

FIG. 17.3 Spectra of normalized magnitudes of terms in the f~ - Vp-tendency equation at hour 48 of a simulation for (a) e = 1, (b) e = 4, and (c) t = 10. Terms are labeled as T (tendency), L (linear), A (adiabatic nonlinear), and D (diabatic) [from Fig. 5 in Errico, 1989c).

1

geostrophic omega-equation to compute profiles of a vertical velocity or horizontal divergence will likely yield unrealistic results if short vertical scales are not filtered.

18. The Description of Gravitational Modes as Oscillators In the previous section, the existence of a nonlinear balance toward which typical models tend was demonstrated for some horizontal and vertical scales. In this section, the reason for this tendency is explained. First, it is demonstrated that the nonlinear prognostic equations for normal mode coefficients which were derived in Section 6 are indeed useful for describing a model's nonlinear geostrophic adjustment process. This is followed by a description of mode behavior in a model during longer time periods sampled from a long simulation. This will serve to relate the forcing and responses of quasi-linear modes.

18.1 Demonstration of short-term behavior The prognostic equation for a normal mode coefficient may be written as dg = -(iA + )g +N(t).

(18.1)

This is the same as (8.1) and (17.1), except that all nonlinear diabatic and adiabatic forcing is denoted as N and a linearized dissipation has been explicitely considered, with e-folding damping rate v. For harmonic forcing of the form N(t) =

Nk

(18.2)

k

the general solution to (18.1) is g(t) =

(g(0)-E RkNk) / kV

e-(+v)t + E RkNkeC-

k

,

(18.3)

where Rk is the response function Rk = (iA-ik

99

+ v)-

.

(18.4)

This is the same result as (8.3), except for consideration of v and the presence of more than one forcing frequency. Essentially, (8.1) has the form of the equation for the amplitude of a simple harmonic oscillator which is damped and harmonically forced. In other words, the behavior of a gravitational mode coefficient is similar in character to that of a spring or pendulum, except that the "external forcing" acting on it may be quite complicated. Actually, (18.1) is exact only in the sense that given N(t), g(t) is determined by that equation. From (18.1) we learn about the effects of the nonlinear forcing of the mode, however we learn nothing regarding the determination of N in general or the possible feedback of g on N and subsequently on g itself (acting through N). Although (18.1) and its solution (18.3) have simple forms, they do indeed describe the behavior of many gravitational waves in models. As examples, we reproduce figures from Errico and Williamson (1988) which show g(t) for some modes in a global NWP forecast begun from a noninitialized analysis. The figures are presented as harmonic dials, which are plots of real vs. imaginary parts of the g as functions of time on the same figure. They are called harmonic dials because a harmonic function will appear as a circle (dial) on the figure. Positive imaginary components are plotted in the bottom half plane so that westward propagation appears as a clockwise sequence of points, as though the wave was observed from above the north pole. Values of g were plotted every one-half hour, and selected times (in hours from the initial time) are indicated in the figures. Dials of four modes are presented in Figures 18.1a-d, corresponding to Figures 2a-d in Errico and Williamson (1988). These are respective dials for: the t = 2, zonal wave number 2, Kelvin mode; the external, largest meridional scale, zonal wave number 10, westward propagating gravitational mode; the t = 4, 2nd largest meridional scale, zonal wavenumber 1, westward propagating gravitational mode; and the t = 4, 12th largest meridional scale, zonal wavenumber 9, westward propagating gravitational mode. In the linearized model used to define them, these modes have respective periods 28.5, 3.4, 27,

100

(c)

Jiba

(d)

60

FIG. 18.1 Harmonic dials for selected modes in a noninitialized CCM forecast beginning from an ECMWF FGGE analysis: (a) I = 2, m = 2 Kelvin mode; (b) I = 1, first-symmetric, m = 10 westward-propagating gravitational mode; (c) t = 4, firstantisymmetric, m= 1 westward-propagating gravitational mode; and (d) I = 4, sixthantisymmetric, m = 9 westward-propagating gravitational mode. The outer dotted circles indicate magnitudes of 0.32, 0.048, 0.12 and 0.12 m s- 1, respectively. Crosses indicate times labeled in hours after start of forecast [from Fig. 2 in Errico and Williamson, 1988].

and 11.4 hours, and the noted directions of propagation refer to those in that linearized model. Behaviors of the presented external modes (Figs. 18.1a-b) appear to fit (18.3) quite well. In particular, both have components with periods close to their linearized values, although the mode in Figure 18.lb is noticeably slowed by the model's semi-implicit scheme (Wiin-Nielsen, 1979; Errico 1984a) which has not been considered in the linearization but which has a large effect on such otherwise fast modes. Both have damping, with the smaller scale mode damped at a faster rate (approximately an amplitude e-folding time of 18 hours). Both appear to have quasi-stationary components based on the evidence that the centers of the near circle during any period appear to be nearly concentric (to a crude approximation) about a point offset from the graphs's origin. More detailed analysis and discussion appears in Errico and Williamson (1988). Behaviors of the internal modes (Figs. 18.1c-d) do not fit (18.3) as well as those of the external modes, in the sense that regular damped propagation is less obvious. There appear to be components with frequencies near the modes' linearized frequencies, but several additional components of similar magnitude and period appear to be present also. This complication may be considered as a consequence of less separation in frequency between the nonlinear forcing and linearized frequency, or it may be considered a consequence of the unrealism of the chosen basic state; i.e., the linearization has failed to distinguish the modes which are more nearly independent in the nonlinear model (see, e.g., Errico, 1983). The modes in Figure 18.1c-d actually may be linear combinations of modes which have behaviors like those in Figure 18.1a-b, which then appear as several regular propagations superimposed. This last comment is based on some fundamental properties of normal-mode analysis. As described in Section 6, we may write

cl = T

(d- bl) , 102

(18.5)

where d represents a set of data, b represents a basic state (such that d - b is a departure - T from that state), and

l

represents a corresponding linear transform from data to a

set of mode coefficients c. The index 1 is used to contrast one basic state from another (index 2) for which (18.5) would appear identically, except with a 2 replacing 1. Since d is independent of the choice of b, a set C2 is related to cl by c2 T =

XT

ll

+ TTlbl - b 2

(18.6)

The operator T lT 1, yields a matrix whose elements v describe the projection of one set of mode structures on another; e.g., for mode q, Vqj Ci

C2q =

(18.7)

plus terms involving bl and b 2 . If basic state 1 yields a set of regularly propagating wave-like departures from that state, meaning that each Cly is associated with a distinct frequency, then according to (18.7), a significantly different basic state 2 will likely yield coefficients, each of which has several frequencies rather than a distinct single one. Therefore, proper interpretation of time series of coefficients requires careful consideration of the choice of basic state used to define the linearized eigenvalue problem.

18.2 Demonstration of long-term behavior The component of g which depends directly on the initial condition g(O) eventually becomes negligible since it is continuously damped.

Therefore, after several damping

periods, the solution (8.3) is well approximated by g(t) =

E

RkNkc At.

(18.8)

k

Although no longer explicit, (18.8) still depends on g(0), but only through its implicit effect on the determination of N(t) and therefore on Nk. Solution 18.8 emphasizes the importance of the response function. The magnitude of the response of g at any frequency Ik is proportional to not only the magnitude of 103

m= 1

112

n= 0

1=1

_u 11fi l I 1l'

10

EG

Pr =

I I I"'

33.0 -'

'

'

(,s

cv

(nI

0o

10

10

-_-

8

0

z

m1*

\

L. LiJ C) 0

z

a. 10

6

0

(n,

LiJ cx

r.

"i m

10

.

I

I

I

-

.

,

,

III

|

I

4

10 2 PERIOD

10

100

(hours)

FIG. 18.2 Response function R for the forcing of eastward propagation (solid) and westward propagation (dashed) for a wave with resonance at a 33 hour period (eastward direction) and linear e-folding damping period of 5 days [from Fig. 16 in Errico, 1989a].

the forcing at that frequency, but also tothe magnitude of the response function at that frequency. The latter magnitude is greater the closer Ak is to A; i.e., the closer the forcing is to resonance. An example of a response function appears in Figure 18.2 for a mode with a linearized model period of 33 hours with eastward propagation and assumed e-folding damping period of 5 days (without damping, R would be infinite at resonance). Harmonic dials of four modes from 16 days near the end of a 1000-day simulation are shown in Figure 18.3a-d. These are taken from Errico (1989a; his Figu

res

pectively)

and may be described as: (a) the fifth largest meridional scale, zonal wavenumber 7, external westward propagating gravitational mode (3.1 hour resonant period); (b) the zonal wavenumber 1, external Kelvin mode (33 hour resonant period); (c) the tenth largest meridional scale, zonal wavenumber 4, t = 5 westward propagating gravitational mode (14 hour resonant period); and (d) the zonal wavenumber 4, t = 4 Kelvin mode (52.2 hour resonant period). Asterisks on the dials are separated by one-day periods. The simulation is sufficiently long so that the presented behaviors are responses to forces acting upon the modes rather than explicit effects of initial conditions. The simulation was performed with the NCAR CCM1. The mode in Figure 18.3a is representative of all fast modes (resonant periods less than 12 hours) in the simulation. It has a strong stationary component (presumably a topographic effect) and a significant quasi-stationary component.

Superimposed are

some small wiggles, which are near resonant gravitational components (as evidenced by Figure 18.4a). The remaining modes (Figures 18b-d) show significant wave-like behavior with periods not much different than the mode's resonant periods. Power spectra of the modes in Figure 18.3a-d are presented respectively in Figures 18.4a-d. These are determined by removing linear trends from the mode coefficients during a 64-day period and performing Fourier analysis on the remainders. Solid lines indicate westward propagating components; dashed lines indicated eastward components. Note that although in the linearized model used to define the modes, each mode

105

1=1 n= 4 -0.08 -

m= 7

, .

.

-0.08:

. .

.

.

i

1

.

I ,

. !! !I .

WG

Pt=

rn.=1

-3.1

n=0 1=

EG

, =33.0

...

.

i .

i

i

i

·i

0.08 _ i i

m

~ ~~~~~~~~ ~~~~~~~~~. .

.

..

~~~~... - 0.08

m= 4

n= 9

1=5

WG

P,= -14.0

m= 4

n

0

1=4

EG

P, =

52.2

FIG. 18.3 Harmonic dials of four modes obtained near the end of a long climate simulation: (top left) third-symmetric, m = 7, e = 1 westward-propagating gravitational mode; (top right) m = 1, e = 1, Kelvin mode; (bottom left) fifth-antisymmetric, m = 4, t =5 westward-propagating gravitational mode; and (bottom right) m = 4, t = 4 Kelvin mode. Asterisks indicate times which are multiples of 24 hours, axes have units m/s, and a-c show behavior for 16 days and d for 64 days [from Fig. 1-4 in Errico, 1989aJ.

10

-4

m-7

n

=

W

,=

-. 1-2

-

1U

n= 0

1=1

EG

P, =

33.0

-4

10

-6

m= 1

a. k.

o,

sa>

.

-6

10

-8

a: a 10 10 w 10 -10

a

ao

10 0-12

10

10

10 PERIOD

_ -2

m= 4

n= 9

10

100

-

0

a, 10

-10

-12

102

(hours)

1=5

WG

10 PERIOD

P,= -14.0

10

10

-8

10

-1

m= 4

n= 0

100

(hours)

1=4

EG

P, =

52.2

-4

10 -66

· .1 ^,0-7

a. -8

= 10 10 0

0. -10

-l

10 11

0

10 PERIOD

101 (hours)

FIG. 18.4 Power spectra of modes corresponding to Fig. 18.3a-d. Solid line is for westward propagating components; dashed line is for eastward propagating components. Some smoothing has been performed at short periods [from Fig. 1-4 in Errico, 1989a].

10°

propagates in a certain direction only, in the nonlinear model, there is no unique association. Rather, the propagation characteristics of a mode depend on the strength and propagation characteristics of its various forcings, but only in one direction does a resonant response exist. The spectra quantitatively reveal the qualitative comments made earlier regarding the dials. In particular, relative peaks are observed near the resonant periods of all modes, but in the case of Figure 18.a, the peak is still small. The relative peak at shorter period which occurs in the component which travels in the direction opposite that of the resonant component is due to resonance of a corresponding time-computational mode produced by the model's semi-implicit time scheme. Note that no band averaging has been performed for these figures, and therefore the energy within each period-band of components is proportional to the area under the curve for that band and a factor which approximately varies as the inverse of the period. Therefore although a peak may be small, it may integrate to a large portion of energy. Clearly, for the internal modes shown, there is substantial energy in the near-resonant bands. This is entirely due to forcing in the model. As further evidenced by Errico (1989a), a significant portion of the forcing which is reponsible for excitation of wave-like behavior is the model's convective heating. In other words, the convection primarily acts to create relatively fast, unbalanced components rather than slow balanced components. The key word here is "relatively", which denotes a comparison between forcing and resonant periods. The tropical internal modes typically have resonant periods of longer than one day. For convective forcing to instead drive balanced components of these modes, it would be required to have the same magnitude and location for periods of several days to weeks, which is exceedingly rare (especially for the Kelvin modes for which months would be required). The analysis in Errico (1989a) demonstrates that the time scale of the modes is in part explained by the time scales of the various forces acting on them. If the time scale of forcing is predominately an advective

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one, and if there is sufficient damping to limit more weakly driven but resonant responses, then the time scale of gravitational modes will be an advective, rather than inertial, one. The same time scale will be observed for the ageostrophic fields of 6 and fo - V 2 p since they physically describe the gravitational modes. This explanation is only partial, because the question of why particular time scales of the dynamic fields are observed has only been shifted to the question of why the forcing has an advective time scale. In particular, since the forcing actually depends on the ageostrophic components themselves, it at least has the potential to have significant high frequency components. Of course, the observation that the time scales of the ageostrophic fields are advective strongly indicates that the forcing acting on them is predominantly advective, as mentioned by Charney (1955). The nature of the forcing acting on the modes will be discussed in the following section.

19. Mode Forcing, Interaction, and the Slow Manifold Charney (1955, 1973) presents brief outlines of why the atmosphere is quasi-geostropic. Essentially, the primary external forcing acting on the atmosphere is solar heating which is characterized by a predominately large space scale (cooling at the poles; heating at the equator) and long time scale (the latitudinal heating contrast is always present to some large degree). Other time and space scales are associated with solar heating (e.g., the time and space scales of clouds passing overhead), but the components at these other scales are either weaker or affect much smaller scales than we are characterizing (essentially we are discussing the synoptic and mesoscales, and medium to large equivalent depths). The response to the primary forcing has the same space and time scales as that forcing, resulting in strong zonal winds aloft in the midlatitudes which have a significant seasonal component but which are otherwise always present. These winds are geostrophically balanced, however they are unstable with respect to small nonzonal perturbations (Charney, 1948; Eady 1948). The perturbations which grow most rapidly are the quasi-geostrophic ones, resulting in propagating quasi-geostrophic waves. These waves are also unstable with respect to

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other quasi-geostrophic perturbations (Lorenz, 1972; Kim 1978), yielding many scales of quasi-balanced fields. If they were also significantly unstable with respect to gravity waves, we would expect to see more gravity waves, so Charney presumed they were not. Also, some dissipation is required to remove any significant energy which may otherwise creep into gravity waves, so that quasi-geostrophic balance is maintained. In the following subsections various aspects of Charney's explanation will be discussed in greater detail. In Section 19.1 the stability of geostrophic waves to ageostrophic disturbances will be discussed. Also, the requirement of dissipation for maintenance of quasi-geostrophic conditions will be discussed.

In Section 19.2 the capability of mode

interactions to result in a "slow" set of solutions will be discussed, along with higher-order balance conditions.

19.1 Stability and Dissipation of Geostrophic Waves Charney's (1955) conclusion that geostrophic waves are stable with respect to gravitywave perturbations was actually conjecture based upon the observations of the lack of gravity wave activity rather than a demonstration of that stability. That geostrophic waves are unstable with respect to other geostrophic waves has been demonstrated by many (Lorenz, 1972; Kim, 1978; Gill, 1974; et al.), however those studies have typical used quasi-geostrophic models which exclude examination of gravity waves. Errico (1981) examined the stability of gravity waves in a periodic two-level f-plane primitive equation model. He used an approach similar to that of Lorenz (1972) and Lin (1980) except he computed interaction coefficients for both gravity and Rossby waves. The basic states were single geostrophic waves (either barotropic or baroclinic), analogous to the rotational modes for the model in Section 3, superimposed on a statically stable

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horizontally uniform state. In his model, these states were stationary solutions to the model equations. Growth rates for quasi-geostrophic wave perturbations were asymptotically proportional to the amplitude of the basic state waves as smaller amplitudes were considered. In other words, for a basic state amplitude of order Rossby number e « 1, or use some estimates of dgn /dt n (n > 1) based on slowly propagating components only. These have been discussed by Baer and Tribbia (1977), Lorenz (1980), Machenhauer (1982), Tribbia (1984), Temperton (1988), and Errico (1989b). Regarding numerical weather prediction, in most cases they change subsequent forecasts very little with respect to other analysis or model errors, because order e n adjustments are extremely small for small E (Rossby number) and large n. However, these higher-order schemes have considerable theoretical interest.

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Part VI. CONCLUSION

20. Summary

This technical note is concluded here with a discussion of several commonly said but incorrect statements regarding nonlinear normal mode initialization: 1) The application of the NNMI balance condition to small horizontal scales is inappropriate. This is false, at least without substantial clarification. The corresponding true statement is that the NNMI balance condition applies to all fast waves as demonstrated in Section 17. These include waves with small horizontal scales which however have large vertical scales, such as external waves. The NNMI balance condition is best conditioned on time rather than space scales. In fact, one particular advantage of NNMI over other initialization schemes is its reference to time and vertical scales. Both must be considered in addition to horizontal scale if proper statements regarding dynamics are to be made. 2) The application of NNMI is inappropriate when gravity waves are important (such as when convective mesoscale systems are present).

The reasoning behind this false

statement was discussed at length in Section 15.1 Here we reiterate that the converse is really true: NNMI is appropriate and necessary when gravity waves are important. The NNMI removes gravity waves which are considered unrealistic but present due to analysis error. If the waves have no impact on a forecast, then it is not necessary to remove them, but if they have an important impact, then any unrealistic waves must be removed, and NNMI is an appropriate procedure for doing that. Realistic waves can be retained or reanalyzed if possible, but certainly significant error-created waves must be removed. 3) The purpose of NNMI is to reduce rms errors in NWP forecasts.

With such

an expectation, NNMI is then maligned if rms scores are not improved. The truth is (Section 15.1), NNMI is designed to effectively remove high-frequency noise due to propagating gravitational waves without substantially affecting the slower and larger amplitude

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behavior of the dynamic fields (except possibly under strong diabatic conditions). After a period of time, other mechanisms remove gravity waves in the absence of initialization. Therefore, after an adjustment period, initialized and noninitialized forecasts should appear very similar. In fact, that is one important test of a correct application of NNMI software. In contrast to its lack of a direct effect, NNMI has significantly improved forecasts by creating better, more noise-free, background (or first guess) fields for use in the analysis of atmospheric data when those fields are produced by previous short-term model forecasts, such as in a forecast, analysis, initialization cycle. Although fast gravity waves have only small dynamic effects on slower components of either gravity waves or rotational modes, slow components can be significantly impacted by affecting their analysis. 4) NNMI is just another "trick" to remove forecast noise, and therefore is a subject of interest only regarding NWP modeling. This is false, as the previous 19 sections have been intended to demonstrate. Rather, the theory of NNMI is as fundamental as quasigeostrophic theory, and in fact explains the basis for that theory (Sections 16-19).

It

elegantly infuses linear concepts into nonlinear contexts, for example regarding nonlinear geostrophic adjustment (Section 8). For these reasons, the theory of NNMI should be part of a basic meteorological curriculum in atmospheric dynamics at the graduate level. 5) With the advent of diabatic NNMI, initialization is a solved problem. The truth is that NNMI has been somewhat hurt by its so successful and easy application. are many remaining problems.

There

Attention has been primarily focused on noise in the

surface pressure field, at the exclusion of attention to internal waves, especially in the tropics. The effects of diabatic processes has been mostly misunderstood (Section 15.3), and modifications to NNMI procedures have been often poorly diagnosed (Section 15.2). There is much work yet to be done, however the greatest potential for normal modes is probably as a diagnostic tool (e.g., as in Section 17) especially where fast and slow, or geostrophic and ageostrophic fields are to be distinguished.

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APPENDIX A: List of Mathematical Symbols The section number where the symbol first appears is indicated in parenthesis. Some symbols which are used in only a single section are excluded from this list. a a A c

Structure functions of the spectral components of normal modes (4.3). A generic field (3). Shorthand for the adiabatic nonlinear terms acting on the mode coefficients (17). The phase speed of a gravity wave (4.3) or a coefficient (complex amplitude) of a generic normal mode (17).

D

Shorthand for the diabatic terms acting on a field or mode, as indicated by a subscript (3.1).

P

The Kronecker delta function (4.1). The total kinetic plus (a quadratic form of) available potential energy, per unit mass (3.2).

E f fo g g' G h H i j L LF

LR Lx Ly I £c

The Coriolis parameter (3). A constant or mean Coriolis parameter (3). The acceleration of gravity (3.1) or a normal mode coefficient (complex amplitude) of a gravitational mode (4.3). The coefficient of a conjugate gravitational mode (4.3). The typical magnitude of a gravitational mode coefficient (7). The phase speed of a gravitational mode with fo negligible (4.3). An equivalent depth of a vertical mode (4.1). The square root of -1 (4). An index for a normal mode (17). A typical length scale (7). A length scale which describes the periodicity of the model domain (3). The Rossby radius of deformation (5.1). A wavelength in the z direction (4.2). A wavelength in the y direction (4.2). An index denoting a vertical mode (4.1). Shorthand for the 3-dimensional linear operator acting on the dynamic fields (4). A linear operator describing the coupling between fields (4). A linear operator (acting on x dependence only) (4).

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0-

A linear operator (acting on y dependence only) (4). a linear operator (acting on a dependence only) (4).

m

A wavenumber in the x direction (4.2)

n

A wavenumber in the y direction (4.2)

N

Shorthand for the nonlinear terms acting on the mode indicated by a subscript, with dependence on modes indicated in parenthesis (6).

N

The magnitude of an oscillating form of N (8). Shorthand for the nonlinear terms acting on a field (6).

p

The hydrostatic pressure (3.1)

Pa

The hydrostatic pressure at the earth's surface (3.1)

p

A mean value of the surface pressure (3.2)

P

A linearized potential vorticity (10)

P

A Fourier transform coefficient of horizontal variations of In (p,/p) (11).

q

Shorthand for the set of scales m, n, t (4.3).

r

The coefficient (complex amplitude) of a rotational mode (4.3).

R

The gas constant for dry air (3.1).

Rk

The response function of a mode to forcing at a frequency

8

The x-dependence of a mode's structure (4).

'

The y-dependence of a mode's structure (4).

S

A factor for simplifying notation (4.3).

t

Time (3.1).

T

Temperature (3.1).

T

A basic state, horizontally uniform temperature (3.2).

TI

A departure of T from T (3.2).

u

The velocity in the x direction (3.1).

fk (18).

A basic state value of u (equal to zero) (3.2). u

A vertical mode coefficient of u (4.1).

u

A Fourier transform coefficient of horizontal variations of u (4.1).

U

A typical magnitude for scaling velocity (7).

v

The velocity in the y direction (3.1).

V

A basic state value of v (equal to zero) (3.2).

v

A vertical mode coefficient of v (4.1).

v

A Fourier transform coefficient of horizontal variations of v (4.2). 120

x y z Z a /3 7f

r 6

A direction which increases eastward (3). A direction which increases northward (3). A function which describes the vertical structure of a mode (4.1). A matrix whose columns are vertical modes of vertically-discrete model (12). A coefficient which describes some mode interactions (6), or a field proportional to the ageostrophic (i.e., linearly unbalanced portion of) vorticity (17.2). A coefficient which describes some mode interactions (6). A coefficient which describes some mode interactions (6). A static stability parameter (3.1). The horizontal velocity divergence (3.1).

C

A scaling parameter or Rossby number (7). The vertical component of vorticity (3.1).

/c

The ratio of the gas constant to the specific heat of dry air for an isobaric process (3.1).

A

The frequency of a normal mode (4.3). Shorthand for the linear operator acting on normal mode coeffients (6). The frequency of a harmonic forcing function (8). A coefficient of linear dissipation (18.1).

A /~ v a r T f>