## 3.1 Modeling and understanding the statistics of weekly averages

The principal mechanism of tropical-extratropical interaction is through diabatically forced Rossby waves. On seasonal and longer scales, tropical diabatic heating is strongly linked to tropical SST; hence one speaks of an "SST-forced" global response as in Chapter 2. On the subseasonal scales of interest here, the SST variability is relatively weak, and its coupling to the heating variability is much less rigid. The heating variability itself is considerable, however, and has a significant extratropical impact. Some of this variability (especially that associated with the MJO) is predictable, and raises the hope that at least some aspects of subseasonal extratropical variability may therefore also be predictable. Unfortunately, for various reasons the simulation and predictability of subseasonal tropical heating variations has thus far proved difficult in general circulation models. This has been a major stumbling block in capitalizing on this source of subseasonal extratropical predictability.

Inspired by the success in Figs 2.1-2.4 of simple empirical predictions of seasonal tropical SST variations and their global impact, we have recently constructed a linear inverse model (LIM) suitable for studies of atmospheric variability and predictability on weekly time scales using global observations of the past 30 years. Notably, it includes tropical diabatic heating as an evolving model variable rather than as an externally specified forcing. It also includes, in effect, the feedback of the extratropical weather systems on the more slowly varying circulation. We have found both of these features to be important contributors to the model's realism.

The model is concerned with the behavior of 7-day running mean
anomalies of extratropical streamfunction and column-averaged tropical
diabatic heating. It assumes that atmospheric states separated by time
lags are related as
**x**(t+) = **G**() **x**(t) + , where **G** is a linear
operator and is
noise. This implies that the zero-lag and time-lag-covariance matrices
of **x** are related as **C**() = **G**()**C**(0). We use this relationship at a particular lag,
say = 5 days, to obtain
**G**(5) from observational estimates of **C**(5) and
**C**(0). We then make another assumption that is at the heart of
the LIM formalism, that distinguishes it from other empirical models,
and that enables one to make dynamically meaningful diagnoses of
direct relevance to modelers. This is that **G**() satisfies the relation
**G**() = exp(**L**), where **L** is a *constant*
linear operator. We use this to obtain **L** from **G**(5), and
having done so, use it again to obtain **G** for all other lags. We
are finally in a position to make forecasts for all lags as
**x**(t+) = **G**() **x**(t). Crucially, having
obtained **L**, we can also diagnose the relative importance of its
elements associated with tropical-extratropical and internal
extratropical interactions. For example, we can use **L** to
estimate what the statistics of extratropical variability would be
without diabatic forcing from the tropics.

Figure 3.1 demonstrates the success of this
model in reproducing the observed variance and 21-day lag covariance
of 7-day running-mean anomalies of 250 mb streamfunction during
northern winter. Note again that we are effectively using the observed
5-day lag covariances to predict the 21-day lag covariances here. The
comparison of the observed and predicted covariances is clearly
encouraging. The right column shows that the effect of tropical
heating is relatively small on the variance but relatively large on
the 21-day lag covariance. This is consistent with our finding that
although tropical heating contributes a relatively small portion of
the extratropical variability, it contributes a large portion of the
*predictable* variability.

**Fig. 3.1**Observed and modeled (using the full LIM and a version of the LIM in which the effects of tropical heating are removed) statistics of weekly 250 hPa streamfunction anomalies.

Forecast skill is an important test of any model. The LIM is better at
forecasting Week 2 anomalies than a dynamical model based on the
linearized baroclinic equations of motion (with many more than the
LIM's 37 degrees of freedom) that is forced with *observed*
tropical heating throughout the forecast. Indeed at Week 2 the LIM's
skill is competitive with NCEP's MRF model with nominally
O(10^{6}) degrees of freedom. The upper panel of Fig. 3.2 shows such a comparison of Week 3 forecast
skill during the winters of 1985/86-1988/89. Other experiments show
that this encouraging forecast performance is not limited to years of
El Niño or La Niña episodes.

**Fig. 3.2**Forecast skill and predictability of weekly averages during winter. Top: Correlation of observed and Week 3 forecasts of upper tropospheric streamfunction anomalies averaged over 52 forecast cases in the winters of 1985/86-1988/89 for (a) LIM and (b) the NCEP MRF. Bottom: Potential predictability limit: forecast lead at which skill (i.e., the correlation of observed and predicted anomalies) drops below 0.5. (c) Determined from the full LIM. (d) Determined from a version of the LIM in which the effects of tropical forcing are removed.

The LIM assumes that the dynamics of extratropical low-frequency
variability are linear, stable, and stochastically forced. The
approximate validity of these assumptions has been demonstrated
through several tests. A potentially limiting aspect of such a stable
linear model with decaying eigenmodes concerns its ability to predict
anomaly growth. We have nevertheless found, through a singular vector
analysis of the model's propagator **G**, that predictable anomaly growth
can and does occur in this dynamical system through constructive modal
interference. Examination of the initial structures associated with
optimal anomaly growth further confirms the importance of tropical
heating anomalies associated with El Niño and La Niña as well as
Madden-Julian oscillation episodes in the predictable dynamics of the
extratropical circulation.

The LIM formalism also allows one to estimate predictability limits in
a straightforward manner. Indeed it allows one to estimate the
expected skill of any *individual* forecast from the strength of
its predicted signal. Given that in many cases the predictable signal
is associated with tropical forcing, one can quantify the effect of
that forcing on extratropical predictability. Our general conclusion
is that without tropical forcing, extratropical weekly averages may be
predictable only about two weeks ahead, but with tropical forcing,
they may be predictable as far as seven weeks ahead. This difference
is highlighted in the lower panel of Fig. 3.2. This suggests that accurate prediction of
tropical diabatic heating, rather than of tropical sea surface
temperatures *per se*, is key to enhancing extratropical
predictability on these time scales.

As mentioned earlier, most current GCMs have difficulty in representing and predicting heating variations on these scales. This is especially true of the NCEP MRF model. We have documented significant deficiencies in the "reanalysis version" of that model in maintaining and propagating MJO-related heating and circulation anomalies. Figure 3.3 shows that forecasts initialized when the MJO is active over the Indian ocean are unable to represent the subsequent eastward propagation of 850-mb zonal wind anomalies; indeed they do not predict propagation at all but a rapid decay. This has been demonstrated to have a negative impact on extratropical forecasts.

**Fig. 3.3**Composite anomalies of 850 mb zonal wind averaged between 5N-15S relative to the maximum of the first EOF of subseasonal tropical OLR anomalies, when MJO activity is maximum over the east Indian ocean. The upper panel is for observed anomalies from Days -14 to +14, where Day 0 refers to the time of maximum EOF coefficient. The lower panel is for the anomalies predicted by the NCEP MRF model, with the mean model error removed.

Figure 3.4 shows that the LIM's forecast skill over the PNA region is comparable to that of the operational MRF ensemble mean, especially in summer. The MRF can represent some phenomena that the LIM cannot, such as nonlinear baroclinic cyclogenesis and blocking. To the extent that these phenomena are predictable, the MRF should have an advantage. This is indeed the case in Week 1. By Week 2, these phenomena become unpredictable; even so, their role in exciting larger scale, slowly evolving structures such as the PNA pattern in Week 1 can contribute to maintaining forecast skill in Week 2. On the other hand, the LIM is much better at predicting subseasonal variations of tropical convection than the MRF, and being an anomaly model, also does not suffer from climate drift by construction. Therefore, it seems likely that the comparable skill of the LIM and the operational MRF models is not arising entirely from the same sources. This is in contrast to the seasonal prediction problem discussed in Chapter 2, in which the comparable skill of GCMs and simple statistical models arises from essentially the same source. To the extent that the sources of Week 2 forecast skill in the statistical and dynamical models are distinct, combining the two forecasts should, in principle, yield forecasts that are superior to either in isolation. Constructing such a combination is currently one of our main priorities.

**Fig. 3.4**Week two forecast skill (as measured by pattern anomaly correlation over the PNA regions) for the operational NCEP MRF ensemble mean (blue curve) and the LIM (red curve) for four winter and summer seasons.