## 1. Introduction

There has been much work in recent years on building models to both simulate, and predict the El Niño–Southern Oscillation (ENSO) phenomenon. These models run the entire spectrum between concise analytical models (e.g., Suarez and Schopf 1988) through large numerical GCMs (e.g., Philander et al. 1992). The simplest models that have achieved some success at prediction are of an intermediate type. In particular the model due to Zebiak and Cane (1987) has demonstrated skill up to a year in advance (Cane et al. 1986). Recent work (Chen et al. 1995) has shown that the predictivecapability of this model can be improved through more refined initialization.

As pointed out by Blumenthal (1991), the Zebiak and Cane model (hereafter ZCM) is not self-adjoint, and therefore it may be possible to find structures in the system that grow faster, at least for a limited time, than the eigenmodes. A series of papers by Farrell (1988b, 1989, and others) point out the value of searching for these optimal growth structures in stability analysis. Also Farrell (1990) and Lacarra and Talagrand (1988) discuss their effect on error growth and predictability. The general idea is that if the structure that grows fastest can be identified, then the error in the system can be minimized by resolving this structure as accurately as possible.

This study develops a linear version of the coupled ocean–atmosphere equatorial model by Battisti (1988), which is a close cousin of the ZCM. This linear ocean–atmosphere model is used to generate the singular vectors of the linear propagator of the system for a fixed time period, *τ.* The first singular vector represents the initial conditions that will grow the fastest over the timeperiod. This optimally growing structure will be called the *τ*-optimal.

This paper is organized as follows. Section 2 briefly describes the physics of the Battisti model and outlines the mathematical solution of the linearized version. The method of linearization and the important equations appear in appendix A. Section 3 outlines the method for finding the optimals: singular value decomposition (SVD). Section 4 demonstrates that the optimization of the thermodynamics and the ocean dynamics can be considered separately under certain conditions. Section 5 examines the various fields of the optimal initial condition and offers physical explanations for their structure. Section 6 reexamines the results of the previous sections for two backgrounds different from the annual average used in the rest of the paper. One of these backgrounds is designed to have a neutral ENSO mode, and the other to have a damped ENSO mode. Section 7 compares the optimals produced here with those calculated by several other studies. Appendix B discusses some of the numerical difficulties that arose in the analysis. Section 8 is the summary.

## 2. Model equation

The Zebiak–Cane model of the equatorial Pacific is well established and has been used successfully to simulate the interannual variability of ENSO (Zebiak and Cane 1987), and to make predictions about the occurrence of El Niño events up to a year in advance (Latif et al. 1994). A model almost identical to the ZCM was developed by Battisti (1988, hereafter B88M) and has been studied extensively (Battisti and Hirst 1989). The model used here is a linearized version of these models called LOAM (Linear Ocean–Atmosphere Model). Throughout this paper LOAM is configured to the Battisti version since a large part of this study is to see if the optimal initial conditions for SST can be computed separately from the other model variables as was done in Chen et al. (1997, hereafter C97) using B88M. The difference between ZCM and B88M are covered in depth by Mantua and Battisti (1995).

The model consists of a prognostically evolving ocean coupled to a diagnostically determined atmosphere. It is an anomaly model, and so all variables are calculated as perturbations on top of user-supplied background fields, which were derived from Rasmusson and Carpenter (1982) and vary from time step to time step with an annual cycle. The details and characteristics of the Battisti model are covered in depth elsewhere (Battisti 1988), and the details of the linearization used here are included in appendix A. Unlike the full Battisti model, LOAM is run with a time invariant, annually averaged background state. The backgrounds for the SST, winds, and upwelling are shown in Fig. 1.

In the case of the Battisti model, a regular ENSO cycle would develop and persist (Battisti 1988). It was believed that the ENSO cycle was the most rapidlygrowing eigenmode of the linear portion of the model equations, while the nonlinearities served primarily to limit the growth of this ENSO mode. A linearized version of the Battisti model was built by Battisti and Hirst (1989), which seemed to confirm this idea. More recently C97, running the full Battisti model, finds that the model is essentially linear for small perturbations over time periods of at least 9 months. Of course, any system can be considered linear for small enough time periods and perturbations, but some of the results of this paper involve longer time periods (greater than 90 days) and some justification that the system is linear this long is required.

*y*by projecting them onto Rossby wave space and discretized in

*x*using finite differencing (see Wakata and Sarachik 1991). The atmosphere and SST are projected onto Hermite function space in

*y*and likewise discretized in

*x*using finite differencing. It is important to note that in the coupled model, the winds and Ekman current are diagnostically determined by the SST; that is, these fields are automatically consistent with the SST. In this paper, “

**T**fields” will refer to SST, winds, and Ekman currents altogether. Likewise, “

**r**fields” will refer to the Rossby (and Kelvin) wave amplitudes, which include the thermocline depth along with its dynamically consistent upper-layer currents. In matrix form the model can be writtenwhere

**r**and

**T**are the

**r**field and

**T**field discretizations, respectively. DefiningGiven a set of initial conditions, the solution to this system is determined at any time by

*φ**t*

**M**

*t*

**φ****M**

**e**

**e**

*x, y*

*iσt*

*i*=

*σ*is a complex number, then (2.1) becomes

**M**

**e**

*iσ*

**e**

**e**

_{j},

*σ*

_{j}) called the eigenvector–eigenvalue pairs, or the eigenmodes. The time-dependent solution to the original problem (2.1) can be written as the sum of these eigenmodeswhere the

*α*

_{j}are the magnitudes of the projection of the initial conditions onto the eigenvectors. In other words,

*α***E**

^{−1}

*φ***is a vector of the initial condition coefficients, and**

*α***E**

From (2.7) it can be seen that the eigenvector associated with the eigenvalue having the largest imaginary part will dominate the solution at sufficiently long times. In general both **e** and *σ* are complex and so, even though a single mode might appear from (2.7) to be growing exponentially without changing shape, this is not necessarily the case. The mode oscillates between the real and imaginary parts of **e** with a frequency determined by the real part of *σ.* For this model the most unstable mode resembles the ENSO cycle and will be referred to as the ENSO mode.

Figure 2 shows the real and imaginary parts of the ENSO mode produced by LOAM. The real part is atthe peak of a warm event, while the imaginary part shows the fields halfway between a warm and cold event. The cold event in this linear model is identical to a warm event but with the signs reversed. The period of this ENSO mode is 1055 days, which is in good agreement with Battisti and Hirst (1989).

## 3. Optimal initial conditions

**r**fields and

**T**fields. How should these two subvectors be combined? This question is addressed in the following way: the norm will be defined as a weighted sum of an L

_{2}norm on each of the the components; that isThis norm will be referred to as the L

_{2w}norm, or as the L

_{2}norm when the weights are at their nominal value of one. The Rossby coefficients,

**r**, were used as a convenient proxy for the thermocline perturbation. The L

_{2}norm of

**r**closely tracks the L

_{2}norm of the thermocline over the ENSO cycle (not shown).

*τ,*the optimal can be computed using the SVD of

**R**

*τ*), the propagator. Singular value decomposition breaks

**R**

*τ*) into three matrices:

**R**

**USV**

^{T}

**U**

**V**

**S**

**V**

**S**

**V**

_{2}norm grows by a factor equal to its singular value, and its final shape is given by the corresponding column of

**U**

**V**

**V**

^{T}

**V**

**I**

**I**

**RV**

**US**

**R**

**v**

_{i}=

*s*

_{i}

**u**

_{i}, where

**v**

_{i}and

**u**

_{i}are the

*i*th columns of

**V**

**U**

*s*

_{i}is the

*i*th diagonal element of

**S**

**R**

*τ*) is the propagator, it takes the field of shape

**v**

_{i}, to a field of shape

**u**

_{i}, and size

*s*

_{i}, at time

*τ.*More details about SVD can be found in Noble and Daniel (1988).

This system can be viewed as having three “sources” of growth: 1) the exponential growth of the ENSO mode, 2) the periodic growth seen leading into the warm and cold events, and 3) the transient growth associated with short-term effects (see Borges and Sardeshmukh 1995). The exponential growth is easy to quantify since it is given by the imaginary part of the ENSO mode eigenvalue. This accounts for the growth *rate* at longertimes, but only accounts for a portion of the growth for periods of time less than a year. Figure 3 shows values of the L_{2} norm over one ENSO period, with the exponential growth removed. Also shown is the portion of the L_{2} norm contributed by the **T** fields, and the portion contributed by the **r** fields. Depending on the values of the weighting factors, w_{1} and w_{2}, the L_{2w} norm will vary, but throughout this paper, unless otherwise stated, w_{1} = w_{2} = 1. With the nondimensionalization used, this makes the contribution to the norm about the same for the **T** fields as for the **r** fields. The graph shows that the norm is cyclic, with a period of one-half the period of the ENSO mode. The two peaks of the L_{2} norm correspond to the peak warm event and the peak cold event in the ENSO cycle. By starting at a minimum point in the cycle, the norm can grow by more than a factor of 2 over a 260-day period.

Owing to the instability of the ENSO mode and the cyclic nature of the L_{2} norm on the ENSO mode, large growth can be achieved without even resorting to adjoints or SVDs. How much more growth can we achieve using our optimization techniques? Figure 4 shows the growth that can be achieved by three different schemes. The bottom curve (scheme 1) shows the maximum growth that can be achieved over *n* days by starting with a pure ENSO mode at the optimal phase, where the instability growth is removed. The second curve (scheme 2) adds in the instability growth, that is, the natural exponential growth of the ENSO mode, while the top curve (scheme 3) shows the growth of the *τ*-optimal. The last curve marked “filtered SVD” shows the growth achieved by the optimal fields presented in the graphs below, which have been filtered to get rid of some grid noise. This filtering is discussed in some detail in appendix B.

For scheme 1, maximum growth of about a factor of 2 occurs for a period of one-quarter the ENSO period. This scheme just starts the ENSO mode somewhere on the upward slope of the L_{2} norm as shown in Fig. 3. If Fig. 4 were extended to show growth periods as longas one-half an ENSO period, it would show that this scheme would be unable to produce growth at that time. Wherever it starts on the L_{2} norm cycle it ends up at the same magnitude half a cycle later. At three-quarters of a cycle it would achieve its maximum growth again, and so on. The second curve shows the amount of growth that the first curve achieves if its natural exponential growth is not suppressed, which is more realistic especially for small perturbations. At 360 days, the instability more than doubles the growth compared to scheme 1, but it has only a small effect for the shorter time periods. The figure also shows that the *τ*-optimal (scheme 3) produces a significant improvement in growth: it is twice as effective as the unstable ENSO mode (scheme 2) for periods as short as 90 days, and almost three times as effective at 1 yr.

Figure 5 shows the growth over time for five different *τ*-optimals. Each of course produces the largest possible value at its target time, but notice that the optimals at 30, 90, and 180 days continue to grow, each peaking at around 300 days. In section 4 it will be shown that the optimals develop quickly (by around 90 days) into the ENSO mode. They just produce this ENSO more efficiently than if one started with a pure ENSO mode. Examination of Fig. 5 shows that, as *τ* increases from 30 to 360 days, the first peak (the mature phase of ENSO) occurs later and is larger. The 450-day optimal, however, peaks before its target day and produces an ENSO mode with smaller amplitude than the 360-day optimal. Is there an optimal *τ*? Since the system is unstable, as *τ* → ∞ the growth of the *τ*-optimals must also go to infinity, but maybe the system has a local maximum.

Figure 6 shows the time and magnitude of the first peak for *τ*-optimals with *τ* ranging from 20 to 1000 days. This graph shows that there is indeed a local maximum at 360 days, and then another at 880 days. The time period, *τ,* where the first local maximum occurswill be called *τ*_{max}. Note from Figs. 5 and 6b that the *τ*_{max}-optimal is the one where the ENSO mode peaks at the target time. This local maximum is essentially the unstable equivalent of the absolute maximum for *τ* found in damped systems (see PS, section 4). But what of the second local maximum? It is outside the scope of this paper, but Thompson (1997) shows that the *τ*-optimals converge to the adjoint well before *τ*_{max}, so the *τ*-optimals repeat cyclically with a period equal to the period of the ENSO mode. The 880-optimal has the same shape as the 360-day optimal (but is of the opposite sign).

## 4. Optimizing on SST versus optimizing on dynamics

**R**

_{22}, where the propagator is divided as[Note that this is not the same as taking the SVD of

**R**

*w*

_{1}= 0 in the L

_{2w}norm. This formulation would leave out the constraint that the ocean dynamics initially be zero and would lead to an ill-posed problem whose solution would be to make the ocean dynamics (

**r**fields) initially infinite while leaving the initial SST zero.] The first singular vector of

**R**

_{22}will be referred to as the

**T**-optimal, while the first singular vector of

**R**

_{11}will be called the ocean dynamics optimal or

**r**-optimal.

**R**

_{22}by running the full Battisti model with small perturbations on the SST field, but found the method too computationally expensive to do for the full set of fields. With LOAM finding the optimals is relatively easy, so the

**T**-optimal can be checked against the SST portion of the full optimal for various time periods. The

**r**-optimal can also be compared to the full optimal. Intuition would suggest that the shape of the

**T**-optimal should become similar to the full optimal for longer times—long enough to give the SST time to interact with the ocean dynamics. From (2.4) and (4.1) an expression for

**R**

_{22}can be found:From this it is clear that

**R**

_{22}will be affected by all parts of the linear operator

**M**

**T**-optimal and

**r**-optimals look like their respective parts of the full optimal, both having a correlation of greater than 85%.

_{1}= w

_{2}= 1, but what if these weights are changed? That the

*shapes*of the subfield optimals are essentially independent of the weighting for

*τ*greater than 60 days indicates that the

**r**-optimal and the

**T**-optimal might be linearly combined to give the full optimal. Because our system has a single growing mode, the ENSO mode, any initial condition will eventually lead to this mode. Let

**β**_{τ}

*t*

**R**

*t*

*α*

_{τ}

*α*_{τ}is a full

*τ*-optimal, and

**β**_{τ}(

*t*) is the field that results after time

*t*has elapsed. Defining

**β**_{ENSO}as the (normalized) shape of the ENSO mode, then for large enough

*t,*where

*α*^{r}

_{τ}

*α*^{T}

_{τ}

**r**-optimal and

**T**-optimal respectively, and the

*σ*are real scalars.

But how long a time is required for this, and will the phase of the ENSO mode be the same for each? Figure 8 gives the correlation coefficient between *β*_{τ}(*t*) and *β*_{ENSO} for the full optimal, as well as the **T**-optimal and **r**-optimal for *τ* equals 90 and 360 days. Figure 9 givesthe phase of the ENSO mode for the same parameters. Figure 8 shows that for periods of time as short as 90 days, (4.4) is approximately true, with a greater than 85% correlation between the optimals after 90-day growth and the ENSO mode. Figure 9 shows that the ENSOs generated by the different optimals are almost in phase, to within 5 degrees by time *τ.* However, these phases do not converge. The explanation can be found in Fig. 3: the norm of the **T** fields and the norm of the **r** fields peak at slightly different times. Since the **T**-optimal is optimizing the **T** fields only, it will try to make an ENSO where the **T** field norm peaks at the chosen time, whereas the **r**-optimal will try to place the ENSO so that the **r** field peaks at the chosen time.

*a*

_{1}and

*a*

_{2}are real scalars. Then the “mixture” of the

**T**-optimal and

**r**-optimal that produces the maximum ENSO can be computed from a simple optimization problem. But first choose the time

*t*=

*τ*large enough so that (i) Eq. (4.4) is valid and (ii) the shapes of the

**T**-optimal and

**r**-optimal are the same as their respective parts of the full optimal. If weighting factors are included, then (4.5) can be used to compute the full optimal:The optimization problem becomes

*a*

_{1}

*σ*

^{r}

_{τ}

*a*

_{2}

*σ*

^{T}

_{τ}

*a*

^{2}

_{1}

*w*

^{2}

_{1}

*a*

^{2}

_{2}

*w*

^{2}

_{2}

^{1/2}

To check this result the full optimal was computed using LOAM and the results are plotted against (4.8) for a wide range of weights. The results found in Fig. 10 show good agreement between the two. Note that this figure is not intended as a practical method for computing the optimal but merely as a demonstration that the approximations leading to Eq. (4.8) are valid over an extreme range of weights.

Equation (4.5) suggests a straightforward way to evaluate the relative importance of the the two different subfields. Since both the **T**-optimal and **r**-optimal produce a field (at time *τ*) of the same shape, the maximum SST of the final field is an adequate description of its size. Table 1 shows the maximum SST produced by the **T**-optimal scaled to have an initial maximum SST of one degree, along with the maximum SST produced byan **r**-optimal with a maximum thermocline perturbation of 1 m. Assuming the worst case, that errors in knowledge about each of the initial fields are in the shape of their respective optimal, what is needed is a judgment about the “value” of the initial fields. A hypothetical forecaster who wanted equal amounts of possible error coming from inaccuracies in the measurement of the thermocline and SST would need to resolve the thermocline field to about 1.2 m if the SST resolution were 0.1° (at 360 days). Figure 11 shows the initial fields and the final fields for the 90-day and 360-day optimals, respectively.

## 5. Physical interpretion of optimals

For the physical explanation of the optimal initial conditions, the 360-day optimal will be examined in more depth. For didactic purposes, the optimal will be divided into four subfields: the symmetric SST, the antisymmetric SST, the symmetric thermocline, and the antisymmetric thermocline. As before, the SST fields also include the implied wind and the Ekman component of the currents, while the thermocline fields include the dynamically consistent geostrophic currents. Figure 12a shows the growth of the L_{2} norm for the full field, starting with each of the four different suboptimals. Figure 12b shows the correlation between the field produced by each of the suboptimals and the field produced by the full optimal. From this figure it can be seen that even though the suboptimals are initially linearly independent, they become colinear as they grow. That is, as time goes on, they all develop into the ENSO mode at the same phase. The total growth at 360 days of the full optimal is then simply the sum of the amplitudes of each of the suboptimals after 360 days. The symmetric part of the **T**-optimal accounts for 68% of thefinal amplitude of the 360-day **T**-optimal, and the antisymmetric part accounts for 32%. For the 360-day **r**-optimal the symmetric and antisymmetric parts account for 80% and 20% of the final amplitude, respectively.

### a. Symmetric part of the **T**-optimal

The symmetric SST portion of the **T**-optimal is shown in Fig. 13a, along with the surface winds that accompany it in 13b. The warm water in the eastern Pacific and cold water in the western Pacific induce strong westerly winds near the equator at around 120°W. The effect of these temperature and wind fields can be broken up into two parts: their effect on the SST and their effect on the ocean dynamics.

Figure 13c shows the derivative of the SST. Most of the heating in the western central Pacific is from two sources: about half of it is due to the *d*(*x, y*) *T* term of Eq. (A.7). The other half is due to relaxation of the anomalous upwelling caused by the westerly surface winds, the *K*_{w}∂*υ*_{1}/∂*y* term of Eq. (A.7). Notice how the SST pattern avoids putting positive SST on the equator in the eastern Pacific. Examining Fig. A3 shows why: the thermal damping term peaks along the equator in the central and eastern Pacific. By putting negative temperature values in the central equatorial Pacific, and avoiding positive values in the eastern Pacific, the optimal achieves a net heating from this term, while still producing the needed westerly winds. In the ENSO cycle the westerly winds are likewise caused by a west to east gradient in the temperature, but the positive SST is right on the equator, reducing its effectiveness. The **T**-optimal thus gets an initial boost from the *d*(*x, y*) *T* term, but by day 30 it becomes a damping factor as the warm anomalies move equatorward.

Figure 13d shows the effect of the initial wind field on the first three symmetric ocean components, that is, the Kelvin wave and Rossby waves one and three. The winds force a downwelling Kelvin signal and upwelling Rossby signals in the east-central Pacific. The downwelling Kelvin signal will grow and propagate eastward, depressing the thermocline and causing heating in the east around the equator. This “secondary” source of heating will eventually provide most of the heating to counteract the damping of the *d*(*x, y*) *T* term. The upwelling Rossby signals grow and propagate westward where they will have relatively little effect on the SST since the heating coefficient *K*_{T} (see Fig. A2) is small there. These upwelling Rossby signals will eventually reflect off the western boundary into an upwelling Kelvin signal, which will switch off the warm ENSO event being built by this optimal, but this will not occur until just after the warm event peaks at the 360-day mark (see Battisti 1988).

### b. Antisymmetric part of the **T**-optimal

The antisymmetric portion of the **T**-optimal is in Figs. 13e,f. The SST field consists of a cold anomaly in the northeast Pacific and a warm anomaly in the southeast Pacific, with a weaker warm anomaly over the central and western Pacific in the north, and its opposite in the south. The primary effect on the winds is to cause strong northerlies near the equator in the east, but it also causes easterlies in the northeast Pacific and westerly winds in the southeast Pacific. These westerlies are not insignificant, with a peak velocity of about half the peak velocity of the westerlies in the symmetric **T**-optimal.

Figure 13g shows the derivative of the SST produced by these antisymmetric fields. Initially only modest overall heating is achieved, with a tendency as in the symmetric case to move the warm anomaly to the equator. About half of this initial heating comes from thebackground meridional currents acting on the large meridional SST gradient in the east, that is, the *υ*_{1}∂*T*/∂*y* term of Eq. (A.7). Most of the rest comes from relaxation of the upwelling as before. The thermal damping term acts primarily to cause the cooling in the southeast. Figure 13h shows the effect of the surface winds on the ocean dynamics, giving the derivative of the Kelvin and first two symmetric Rossby waves. The effect is almost identical to that in the symmetric case, but with about one-third the amplitude. As before, these perturbations are self-reenforcing and promote ENSO growth.

### c. Symmetric part of the **r**-optimal

The initial thermocline and zonal current for the symmetric 360-day **r**-optimal are shown in Figs. 14a,b. The thermocline is the more important of these two fields and consists of a trough extending the length of the Pacific basin along the equator, deepening at the eastern boundary. The deep thermocline in the eastern equatorial Pacific causes heating there, as shown in Fig. 14c, which gives the rate of change of the SST. Almost all of this heating is due to the *K*_{T}*h* term in Eq. (A7), which acts in the eastern equatorial Pacific (see Fig. A2). As before, this heating causes westerly winds in the eastern and central equatorial Pacific, which will further deepen the thermocline there.

Why does the optimal not ignore the rest of the basin and maximally deepen the thermocline in the east to maximize the SST heating? A look at the time evolution of the ocean dynamics shows what is going on. Figure 15 gives the magnitude of the Kelvin wave, K, and of the first two symmetric Rossby waves, R1 and R3, at 50-day intervals starting with the symmetric **r**-optimal. Here K has nearly a constant positive magnitude across the basin, as does R1, except R1 has an extra “kick” near the eastern boundary. This kick is effective in R1 since the thermocline profile of this Rossby wave peaks slightly off the equator near where the coefficient *K*_{T} peaks. However, it is also a relatively “expensive” feature, since the SST and winds that it stimulates work against it. Examining Figs. 15a–c, it can be seen that this initially positive peak gets pushed down as it propagates westward and is, in fact, negative by day 100. Remember that the L_{2} norm is being maximized here, so quantities that increase in magnitude contribute to the growth, but any quantity that shrinks is expensive. Looking at Fig. 15, it can be seen that R3 starts out negative in the east and grows under the influence of the winds. The thermocline structure of R3 is such that its negative value produces a modest positive perturbation to the thermocline at the equator (and therefore heating), but this modest heating is inexpensive since the winds reinforce rather than oppose R3.

The initial Kelvin profile reflects the same type of trade-off as the R1 profile. An unforced Kelvin signal propagates eastward and then reflects off the eastern boundary as westward-bound Rossby signals. If the initial K profile were to have all of its amplitude near the eastern boundary, it would indeed produce a larger SST in the east, but the reflected Rossby signals it generated would be squashed by the westerly winds caused by the SST. Instead, K feeds positive amplitude from the west over the course of the 360 days. At first it does this on its own, but later uses the the positive amplitude that the Rossby waves start out with in the western Pacific, which reflect off the western boundary into the Kelvin wave (see Fig. 15).

### d. Antisymmetric part of the **r**-optimal

The antisymmetric part of the **r**-optimal is not so complicated. Since the antisymmetric Rossby waves are not strongly influenced by the growing warm anomaly in the east and since they do not reflect off either boundary, heating is their primary effect. Figures 14d and 14e show the initial thermocline displacement and the accompanying zonal currents. The heating is shown in Fig. 14f. About half of the heating is caused by the thermocline suppression at about 5°S, 100°W, while the remainder comes from downwelling and the surface currents that advect along mean temperature gradients (not shown).

### e. The subfields recombined

Since the system is linear and *τ* is sufficiently large (see Fig. 8b), these four subfields are simply added together to form the complete initial conditions. The evolution of the full initial conditions is just the sum of the fields produced by the separately evolving subfields. Since the subfields are initially linearly independent, the initial size of the L_{2} norm is the square root of the sum of the squares of their individual amplitudes. By the end of the period *τ* the subfields are collinear, so the L_{2} norm of the complete field is just the sum of the L_{2} norms of the subfields.

## 6. Neutral and damped cases

The preceding sections show that for an annually averaged background, the Battisti model has the following properties: (i) that the initial growth rates of the *τ*-optimals are significantly greater than the growth rate of the ENSO mode (Fig. 4), (ii) there exists a *τ*_{max} (at 360 days) whose optimal out performs those around it (Figs. 5 and 6), (iii) that the optimals quickly develop into the ENSO mode (Fig. 8), and (iv) that **r**-optimals and **T**-optimals have the same shape as their respective parts of the full optimal (Fig. 7). The last two of these properties seem as though they may result from the instability of the ENSO mode when using the annual-averaged background.

For comparison, two other background fields were tested: one which produced a nearly neutral ENSO mode and one for which the ENSO mode is damped. Theannual average background fields were obtained by averaging the monthly fields from Battisti (1988). Table 2 shows the growth rate and period of the ENSO mode using the background fields for each month individually. For each month the ENSO mode is also the most unstable (or least damped) eigenmode. The damped ENSO was constructed from a mean field that is the average of three stable months, that is, from a perpetual spring. The neutral ENSO background was constructed as the average over five consecutive months that included December through March, plus some percentage of both November and April. The percentage of the unstable November and stable April was varied until a (nearly) neutral ENSO was obtained. Table 2 also shows thegrowth and period of the ENSO mode for these two cases plus for the nominal background. Note that the values in Table 2 are consistent with Fig. 13 from Battisti and Hirst (1989), which shows the results of a similar calculation.

The neutral and damped backgrounds were then run though the same tests as the annual average. Qualitatively the optimals produced by these backgrounds have the same characteristics as listed above, though there are quantitative differences. The most important similarity is that the patterns of the optimals are alike for all three cases, while the most important difference is that the transient growth rates are smaller for the neutral and damped cases. Figure 16 shows the 240-day optimalfor the neutral case, both initially and after 240 days. A 240-day period was picked since it was the value of *τ*_{max} for this case. It looks very similar to Fig. 11c, the *τ*_{max}-optimal for the nominal case. However, it producesless growth, with a 4.9-fold increase in the L_{2} norm. For comparison, the 240-day optimal for the nominal case grows by a factor of 9 (see Fig. 4). The damped case, at its *τ*_{max} of 180 days, produces only a threefold growth in the L_{2} norm, compared to about 7.5 for the nominal case 180-day optimal. But the shape of the damped optimal (shown in Fig. 17) is still very similar to the others.

The figures describing the four properties listed above will not be reproduced here for the two new cases, but they will be described. (i) While the neutral and damped cases do not produce as much transient growth as the nominal case, they do significantly outperform their respective ENSO modes. Even the system that has no growing modes has an optimal that can produce three-fold growth in 180 days. (ii) Graphs of the type in Fig. 6 were produced for the neutral and damped cases, which were qualitatively similar, but with smaller peak growths. The most effective growth periods were 240 days and 180 days for the neutral and damped cased, respectively. (iii) In all cases the optimals grow into the ENSO mode, Figs. 16b and 17b show their respective optimals after *τ*_{max} days of growth. Each is at a peak warm event of ENSO. (iv) Like the nominal case, the **r**-optimal and **T**-optimal can be computed independently for large enough *τ.* The time period for which *τ* is “large enough” becomes larger. The 85% correlation from Fig. 7 is reached for the nominal case by *τ* = 60 days, whereas for the neutral and damped cases it takes 90 and 120 days, respectively.

The experiments with different background states indicate that the instability of the ENSO mode is not the primary reason that the optimals are able to produce growth. That the qualitative results are not sensitive to details of the background state also helps to justify theuse of an annual-averaged background instead of a changing background. The neutral and damped cases also demonstrate another idea: that it is possible to grow ENSO-like warm and cold events even if the system is stable. The ZCM and B88M imply that the ENSO peak events are one phase of an unstable, interannual coupled basin mode, but others (in particular PS) have suggested that the ENSO peak events are instead the transient growth caused by stochastic forcing in a stable system. The implications of the results of this section to PS are discussed in section 7d.

## 7. Comparison with other studies

In this section the results of three other recent papers will be compared with the results of this study. Two papers are devoted to finding the optimal initial conditions for the Battisti model (B88M) or the Zebiak–Cane model (ZCM). These two studies, Chen et al. (1997, hereafter C97) and Xue et al. (1997a,b, hereafter XCZ), use different methods for calculating the optimals and explore different aspects of the problem but nevertheless produce optimals that can be directly compared to those here. The third paper, Penland and Sardeshmukh (1995, hereafter PS), finds the optimals of a Markov model created from SST observations. The model used by PS is different enough from B88M that one should not expect the optimals to be the same; however, it is interesting to make comparisons with an observational study. This is not meant to be an exhaustive study of all the literature on optimal transient growth in ENSO models. In particular, Moore and Kleeman (1997a,b) compute the optimals for an alternate intermediate coupled equatorial Pacific model of Kleeman (1993).

### a. Chen et al.

C97 takes B88M and calculates the linear tangent model using the method suggested by Lorenz (1965). The model is first run over a set period of time, *τ,* to establish a reference trajectory. The model is then run repeatedly, each time perturbing the initial field a small amount on a single element of the state vector. The change in the reference trajectory for each perturbation is recorded, and then the (linear tangent) propagator, **R***τ*), can be approximated. Due to computational restrictions, the propagator was restricted to the SST and so is equivalent to the **R**_{22} matrix used here. Since C97 ran the full model, dynamic consistency of the initial conditions is not required. However, in this study, where a perturbation to the SST field automatically causes the associated winds to exist, the full model will take time to spin up the winds. Theoretically this spinup time is short. Questions about the effect of computing the **T**-optimal independently of the **r**-optimal, as is done in C97, was the original impetus for this study. A major difference between the two papers is that C97 computedthe propagator with changing background states, whereas here a constant background state was used.

Despite these differences there is good agreement between the SST part of *τ*-optimals in this paper and the **T**-optimals from C97. Figure 2 from C97 shows the 180-day optimal starting at four different months. All four of these optimals have the east–west dipole pattern across the basin and the north–south dipole in the eastern Pacific, similar to Fig. B1. The warm anomaly in the southeast has a maximum about twice the size of the minimum of the broad cool anomaly in the western central Pacific, similar to Fig. B1. The main difference is the meridional extent of the warm anomaly. C97 shows it extending out to at least 11°S, whereas in Fig. B1 it extends to about 18°S. Because of computational limits, C97 did not look beyond about 11° from the equator, so the optimals are consistent, at least as far as they are shown. Some meridionally extended runs done by the authors of C97 show that the warm water does extend farther south, but these were not included in their paper (Y.-Q. Chen 1996, personal communication).

The growth rates from C97 can also be compared, at least approximately, to this model’s growth rates. C97 computes the **T**-optimals with an annual-averaged background for only a few *τ,* but the average growth rates can also be estimated from C97’s Fig. 1. Table 3 gives these growth rate estimates, alongside the growth rates for same period **T**-optimals as computed by LOAM. The growth rates for the same *τ* agree to within 20%.

### b. Xue et al.

XCZ took a different approach to reducing the dimension of the problem while using a similar approach to C97 for linearization. XCZ first created a reduced EOF space from a series of 2-yr hindcasts using the ZCM. The tangent linear operator was then generated by perturbing the model with each of these EOFs. The optimal initial conditions were then taken as the first singular vector of these propagators. Like C97, XCZ created propagators that reflected changing background states—but likewise found that it did not make much difference in the shape of the optimal fields. Unlike C97, XCZ’s optimals are for the full set of fields.

Despite using a slightly different model than thisstudy (ZCM vs B88M), and despite using a very different analysis technique, the optimals produced by XCZ are similar to those found here. Figure 6 from XCZ shows a composite average of the first singular vectors of 12 6-month propagators, each starting at a different month of the year. This is roughly the equivalent of the 180-day optimal with an annually averaged background, and so will be compared to Fig. B1. The SST field has a qualitatively similar structure with a warm anomaly in the southeast Pacific and a cold anomaly in the western central Pacific. On the other hand, XCZ’s warm anomaly is more equatorially confined, extending down to only about 5°S, as opposed to 18°S here. Also, the cold anomaly is smaller and more intense: it has roughly the same magnitude as the warm anomaly and is confined to the area between the date line and 130°W instead of extending all the way to the western boundary. XCZ’s thermocline is also similar to Fig. B1, with a valley along the equator from about 5°N to 5°S crossing the entire basin. Similar to Fig. B1, the valley narrows slightly at 120°W and is broader at the eastern and western boundaries.

XCZ also does an experiment where an SST-only optimal is produced in an attempt to reproduce the fields from C97. The optimal produced (see XCZ Fig. 19) more closely resembles both the **T**-optimals of C97 and the **T**-optimal calculated here. The warm anomaly extends farther south to at least 11°S. Also, the cold anomaly in the central Pacific extends almost to the western boundary. The average of the growth rates of XCZ’s 180-day **T**-optimals from XCZ’s Fig. 19 are shown in Table 3 for comparison against the 180-day **T**-optimal growth as calculated by LOAM. XCZ’s growth rate is about 40% larger, but considering the growth rate’s sensitivity to different background conditions, this should be considered good agreement.

### c. Penland and Sardeshmukh

Penland and Sardeshmukh advance the hypothesis that the ENSO cycle is not a linearly unstable system that is constrained in size by nonlinearities, as in the freely evolving ENSO models of B88M and ZCM. Instead, PS suggest that the Pacific basin is linearly stable and that the warm and cold events are transients that grow when an optimal structure is randomly created by noise. To demonstrate that this is possible, PS takes 40 yr of SST observations for the tropical Indo–Pacific basin and builds a linear Markov model. Because of the nature of this empirical modeling, all eigenmodes of the linear Markov model are damped. Penland and Sardeshmukh then find the singular vectors of the model for various time periods, *τ.*

The PS model is obviously quite different from LOAM. Even if B88M were an exact model of the real Pacific SST, the PS model would not necessarily produce the same optimals as LOAM due to its inclusion of the Indian Ocean. However, since PS’s model is thefirst-order autoregression of the 7-month transition, their 7-month optimals are their model’s equivalent of LOAM’s 7-month **T**-optimals.

Like this study, PS found that the shape of the *τ*-optimal was not overly sensitive to *τ,* at least in the range of 3–15 months. For comparison note the similarity between the **T** fields of the 90-day and 360-day optimals in Fig. 11 of this study. PS found the maximum growth occurred for the 7-month optimal (i.e., *τ*_{max} = 7 months). Figure 6a of PS shows their 7-month optimal. Like optimals found here it has a large warm anomaly in the southeast Pacific forming a north–south dipole across the equator in the east. It also displays the east–west dipole along the equator, which produces the westerly wind in the eastern Pacific. However, PS’s optimal also has features not found in any of the studies discussed here. On the equator in the western Pacific is a second large warm anomaly, and then farther west the Indian Ocean is cold. Also a band of warm water extends from the warm anomaly in the western Pacific to Baja California. Figure 6b of PS shows the state of the SST after 7 months of growth starting from the optimal. This graph shows the SST at its peak and consists of a large warm anomaly in the eastern equatorial Pacific, similar to an ENSO warm event. PS’s Fig. 6b also shows that the Indian Ocean has changed from cold to warm over the 7 months.

The results of the neutral and damped background experiments explained in section 6 are not inconsistent with the PS hypothesis that ENSO warm and cold events could be the product of transient growth in a stable system. PS find an optimal that grows an ENSO-like warm event in 7 months in a damped system; here optimals are found that grow ENSO-like warm events in 6 and 8 months for damped and neutral system, respectively. The growth here is somewhat less: the PS optimal achieves a fivefold increase in the L_{2} norm (of SST), whereas the **T**-optimals for the damped and neutral cases here achieve twofold and threefold increases in the L_{2} norm of the **T** fields, respectively. However, the nominal case of this study would also support an alternate hypothesis: that the Pacific has a linearly unstable periodic mode limited in size by nonlinearities. In this hypothesis the optimals, induced randomly by noise, would either cancel or intensify the otherwise regular warm and cold events.

## 8. Summary and conclusions

The properties of the optimal growth structure for a linearized version of the Battisti coupled model of the equatorial Pacific with a constant background state were explored. It was found that the first singular vector of the linear propagator (the *τ*-optimal) grows significantly faster than the most unstable eigenmode (the ENSO mode) of the system. It was shown that the *τ*-optimals grow rapidly into the ENSO mode (around 90 days). For *τ* less than 500 days, a local maximum, *τ*_{max}, wasfound that outperformed the *τ*-optimals around it and has the characteristic that the ENSO mode thus produced peaked at time *τ.* This *τ*_{max} is 360 days for the annually averaged background case, 240 days for a 5-month average (neutral ENSO) case, and 180 days for a 3-month average over the dynamically stable months of February through April (damped ENSO). While the growth rates (singular values) for these different background fields are very different, the shape of the optimal fields are quite similar.

Experiments were run to compare the shape of the optimals produced by looking at (i) the propagator of the **r** fields alone, or **r**-optimals; (ii) the propagator of **T** fields alone, or **T**-optimals; and (iii) the propagator of both fields, or the full optimal. It was demonstrated that when optimizing for a sufficiently long period, the suboptimals take on the same shape as their respective parts of the full optimal. The value of this “sufficiently long period” changes depending on the background fields. The nominal case achieved 85% correlation between both of the suboptimals and the full optimal by *τ* = 60 days, whereas it took 90 days and 120 days for the neutral and damped cases, respectively. It was shown that for a given *τ,* initializing the linear model with either one of the suboptimals by itself produced an ENSO mode of the same phase as the full optimal. For the nominal case it was shown that the **T**-optimal pattern with an amplitude of 0.1 degree is about as efficient as the **r**-optimal pattern with a 1.2-m amplitude at generating a mature ENSO.

Finally, the optimal initial conditions calculated here were compared with optimals from the work of Xue et al. (1997a,b), Chen et al. (1997), and Penland and Sardeshmukh (1995). PS used a very different model, computing a Markov model from SST observations, but still their optimals are at least qualitatively similar to those calculated here. C97 used the same model as this studybut linearized in a different manner, and then looked only at the SST. Nevertheless, the **T**-optimals produced here are almost the same as the SST optimals of C97. Xue used the ZCM, a close cousin to B88M. Even though they employed a very different technique to compute the linear tangent propagator, their optimals and their growth rates are in good agreement with those produced here.

## Acknowledgments

This work was supported by a grant for the NOAA/Office of Global Programs to the Stanley P. Hayes Center for the University of Washington. Many thanks for the long-term support of Ed Sarachik, David Battisti, and Margaret Blacklab. Thanks also to Ying-Quei Chen and Nate Mantua for explanations of the intricacies of the Zebiak–Cane model and the Battisti model. Many useful comments came from Yan Xue and two anonymous reviewers.

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## APPENDIX A

### Linear Ocean–Atmosphere Model Equations

The model consists of diagnostically determined atmosphere coupled to an ocean with a prognostic upper layer with an embedded diagnostically determined surface layer. The atmosphere is driven by the ocean via the temperature of the surface layer, and the ocean is driven by the atmosphere through surface wind stress. The model equation independent variables are *x, y,* and *t* representing the zonal displacement, meridional displacement, and time, respectively. A list of all constants can be found in Table A1.

*U, V*), the geopotential height (

*φ*), and the surface-layer temperature (

*T*). The thermodynamic equation iswhereis the linearization of the convergence feedback loop in which

*H*(·) is the Heaviside function.

*u, υ*) and thermocline are forced by the surface winds according towhere

The model also includes a surface (Ekman) layer driven diagnostically by the winds (see Zebiak and Cane 1987).

*T*

_{x}

*u*

_{1}and

*T*

_{y}

*υ*

_{1}give the advection of anomalous surface currents on the background SST gradient, while

*u*

_{1}∂

*T*/∂

*x*and

*υ*

_{1}∂

*T*/∂

*y*give the advection by the background surface currents on the anomalous SST. Figure A4 shows the annually averaged zonal surface current,

*u*

_{1}. The term

*K*

_{w}(∂

*u*

_{1}/∂

*x*+ ∂

*υ*

_{1}/∂

*y*) gives the cooling due to anomalous upwelling with the coefficient

*K*

_{w}being determined by

*K*

_{w}(

*x, y*) = 0.75

*H*

_{1}

*T*

_{z}

*H*(

*w*

_{1}), where

*T*

_{z}and

*w*

_{1}are the background vertical temperature gradient and upwelling, and the operator

*H*(·) is the Heaviside function. The coefficient

*K*

_{w}is shown in Fig. A1. The cooling due to background upwelling acting on the anomalous vertical temperature gradient is given by 0.75

*H*(

*w*

_{1})

*w*

_{1}·(

*T*−

*T*

^{′}

_{s}

*h*)/

*H*

_{1}, whereis a parameterization of the subsurface temperature for a given background thermocline displacement,

*h*

*K*

_{T}and

*d.*The coefficient

*K*

_{T}gives the heating due to thermocline changes and is given by

*K*

_{T}(

*x, y*) = 0.75/

*H*

_{1}·

*H*(

*w*

_{1})

*w*

_{1}·

*T*

^{′}

_{s}

*h*

*d*(

*x, y*) includes a thermal damping term as well as the cooling due to background upwelling, and is given by

*d*

*x, y*

*α*

_{s}

*H*

_{1}

*H*

*w*

_{1}

*w*

*d*for the annually averaged case.

## APPENDIX B

### Numerical Concerns

As stated in section 2, the model equations were solved by finite differencing in *x* and by a spectral method in *y.* For all of the results in this paper, the LOAM ocean used 15 grid points in *x,* eight Hermite functions for the SST in *y,* and nine Rossby waves for the ocean dynamics in *y.* (LOAM automatically uses one less function in SST than ocean dynamics to avoid a truncation problem.) The atmosphere automatically uses the same grid size in *x* as the ocean but extends around the earth. In *y,* the atmosphere was projected onto 14 Hermite functions.

To test the resolution, runs where made at double the resolution in *x,* with no substantial change in the shape of the ENSO mode or its optimals. The results of the spectral resolution test in *y* were a little more complicated. The model was able to simulate ENSO with as few as two waves: the Kelvin and gravest Rossby. The optimals produced from this minimalist ENSO were of course symmetric since only symmetric functions were used. By adding just the first antisymmetric mode to the ocean and SST expansions, the optimal fields thus produced took on all the basic characteristics of higher resolution runs. However, the ENSO mode produced at this coarse resolution is only slightly unstable because it only has one symmetric Rossby mode to carry energyfrom east to west. When the Kelvin wave reflects off the eastern boundary, it generates the westward traveling Rossby waves with only about half the amplitude being carried by the first Rossby mode (Clarke 1983). The fewer the number of Rossby modes used, the “leakier” the eastern boundary becomes. The number of Rossby modes used in the model runs was picked in part to ensure that the growth rate of the ENSO mode converged. The other condition would be that the optimals have converged. Increasing the number of Rossby waves from 9 to 19 increased the growth of the 360-day optimal by about 20% but increased the run time too much to be practical for all the parameter studies done here. However, a single high resolution (19 Rossby mode) 180-day optimal is presented in Fig. B1, which shows that the primary effect is to extend poleward the meridional extent of the warm water in the SE Pacific, relative to that in Fig. 11c.

The optimals produced by the model consistently had one numerical resolution problem. Examining the background field for the zonal surface currents (Fig. A4), a strong eastward current can be seen, peaking in magnitude at 5°N on the western boundary. This current acts on the *gradient* of the anomalous SST field in the thermal equation (A.7) to produce change in the temperature field. In trying to maximize the gradient of SST right on the boundary, the optimals force a two delta-*x* wave, which propagates across the whole basin. Doubling or tripling the number of grid points in *x* does not help: the grid noise remains. So a simple 1–2–1 filter was applied to smooth the optimals. This removed the noise but also degraded their performance slightly. Figure 4 gives curves for the growth of both the filtered and unfiltered optimals. It shows that the degradation in performance is at most 15%. Throughout the paper all pictures of the optimal fields are filtered, but all the parameter studies use the unfiltered fields. This noise does not appear in the ENSO mode, only in the optimals.

This noise may be an indication of a real phenomenon but, if so, it is on a smaller scale than this model can resolve. Furthermore, the equations are long wavelength approximations, and so do not model small-scale processes. Therefore, small-scale effects that appear are probably not physical. In fact, the background field that causes the problem is not valid on short spacial scales:it has current going into and out of the western boundary. The long wavelength boundary condition for the western boundary is that the integral of the zonal velocity over the boundary is zero (Cane and Sarachik 1976) and is met by this background. However, the more general boundary condition, that the zonal current be zero everywhere, is not met. For this reason it was felt that this phenomenon could be safely ignored.

Size of the SST field grown over *τ* days started from either the **T**-optimal (first column) or the **r**-optimal (second column) of unit size. All fields sizes are measured as the maximum of the field.

Period and growth rate of the ENSO mode for various background fields. Growth rates are *e*-folding times. The first 12 are monthly averaged fields. The annual-average field is the nominal background used throughout this study. The damped case is Feb–Apr averaged. The neutral case is a 5-month average starting with 7% of Nov though 93% of Apr. For the neutral case to be truly neutral, it would need an infinite period of growth; in fact, it is slightly damped with an *e*-folding time of 23 centuries.

Comparison of average growth rates of SST optimals from this study, Chen et al. (1995), and Xue et al. (1997a). Unmarked growth rates are for an annually averaged background state. The growth rates in angle brackets are the average growth rate for changing background states taken over runs starting at each month of the year; NA stands for not available.

Table A1. Values of model equation constants.

^{}

* Joint Institute for the Study of the Atmosphere and Ocean Contribution Number 372.