Predicting ENSO

1. WEATHER FORECASTS vs. CLIMATE FORECASTS

For practical purposes climate prediction is different from weather prediction in important ways. First, the adjustment time of the atmosphere to a thermal or dynamical perturbation is very fast (days to a few months) compared to that of the ocean, land surface or land ice (seasons to millennia). Hence, in making weather forecasts you can fix the water temperature and ice extent to be that observed. The accuracy of the weather forecast depends on how good your numerical model is and (mostly) the quality of the model initialization; i.e., how closely you are to reality at the time you start the forecast. The limit for skillful weather prediction is a few days to a week.

Climate forecasts take advantage of the fact that there are media (ocean/land/cryosphere) that have larger thermal and dynamical interia than the overlying atmosphere, and that the atmosphere responds in a statistical sense to these slower changes in lower boundary conditions. Of course, climate is the aggregate statistics of weather. Hence, by definition a climate forecast is a forecast of the coupled atmosphere/ocean/land/cryosphere system. These are most commonly made by coupling an Atmosphere General Circulation Model (AGCM) to a numerical model of the slower media (typically a land surface plus ocean model). It appears that the limit of predictability of a climate forecast is a few seasons to a year, and this limit is mainly determined by the efficacy of the slow component of the coupled model and the AGCM, and by how accurately the slow component is initialized to the real world at the time of the forecast is started; the climate forecast is insensitive to the state of the atmosphere at initialization time.

2. ENSO FORECASTS

El Nino/Southern Oscillation (ENSO) is a climate phenomenon that is due to coupling between the atmosphere and ocean in the tropical Pacific. Important for the evolution of the atmosphere is the distribution of the Sea Surface Temperature (SST) that creates near surface pressure gradients in the atmosphere (via atmosphere-ocean heat exchange, hydrostatic balance, and the ideal gas law) and hence drives the winds. Important for the ocean is the surface wind stress, which drives ocean currents and upwelling (especially along the equator) and hence affects strongly the SST.

The time scale of ENSO (interannual) results from a competition of processes with comparable time scales (6-10 months): the dynamical adjustment time of the upper ocean in the tropics to a change in the wind stress; the time for the upper ocean and atmosphere to come to a thermodynamic equilibrium. The result is a interannual climate phenomenon that occurs every 2-7 years, with an average time between peak events (eg., El Nino) of about four years. ENSO creates global scale climate changes via atmospheric teleconnections.

A forecast of the state of ENSO over the next year is an example of a climate forecast. The forecast accuracy will depend on the quality of the AGCM, the ocean model (often an ocean GCM), the representation of the energy and moisture transfer between the two media, and the quality of the initialization of the ocean. (the state of the ocean at initialization time is a result of the integrated effects of the wind forcing over the past few years). This is very different set of criteria from the criteria for a good weather forecast (cf, section 1).

3. A DIADACTIC MODEL OF ENSO

A simple diadactic model of ENSO can be used to illustrate the basic methodology used to make a climate forecast, and the type of products that result from a climate forecast. The essence of this toy model of ENSO is as follows. ENSO is usually described as the departure (anomaly) of the coupled atmosphere/ocean system from the average climatological annual cycle. For our puposes, the state of ENSO is defined using two quantities: the SST anomalies averaged near the equator in the eastern half of the tropical Pacific (Nino3.4, T) and the thermocline depth anomaly, h, in the eastern equatorial Pacific (the latter tells us something about the dynamical state of the ocean and is important for the evolution of ENSO). In this model, we will assume that the SST changes slowly compared to the time scale of the synoptic waves (weather) in the atmosphere, so that a portion of the atmospheric variability is affected by the changing SST (observations indicate about half the variance in the surface wind in the tropics is directly related to variance in SST). Hence, the slow changes in the atmosphere (e.g., the Southern Oscillation) are implicit in T. The evolution of the toy model of ENSO can thus be written as:

dZ/dt = R Z,

where Z = {T, h}, and t is time. The matrix R contains the ocean dynamics and the part of the atmospheric variability that is affected by the changing SST. Of course, the atmosphere will still have weather events (with time and space scales of days and <2000km, respectively) that are largely independent of the (small) changes in SST. These weather features provide a stochastic forcing on the ocean that will have some affect on the SST, and hence they will affect the evolution of ENSO. Since this weather is unpredictable and largely independent of small changes in SST, we add it as a stochastic forcing to the ocean. Since observations suggest the ENSO phenomenon is nearly linear, the simple model is actually solved in discrete time steps:

Z(t+tau) = M Z(t) + N(t),

where N is the stochastic forcing.

4. ILLUSTRATION OF ENSO FORECASTING USING THE TOY MODEL

4a) Running the model

The MATLAB script “enso.m” contains a toy model of ENSO.

1)Run the model by typing “enso” in MATLAB. You will be asked how long you want to integrate the model; try 50 years. If we define an El Nino event as a positive SST anomaly that is 1.0 standard deviations above the mean (the red horizontal line in the plot) that lasts for 6 months, estimate the average time between ENSO events.

4b) Estimating the forecast skill of the coupled system

Forecast skill depends on how good your model is (we will assume it is perfect), the forecast lead time, and how well you have initialized the model. Lets examine each of these separately. First, lets make lots of forecasts and calculate some typical measures of skill.

2)Create a time series of the Truth by running the model for again, this time for 300 years.

3)First, assume we have perfect initial conditions for every forecast (chose perfect=1), and lets make forecasts for 40 months. The code will run and create a 40-month forecast, starting every 10 days for 300 years. [That is 300*(360/10)=10,800 forecasts!] Each forecast will start with the exact observed initial conditions. The n-month forecast skill is measured by creating a time series of all the n-month forecast Nino3.4 index an comparing that time series to the Truth using linear correlation (Figure 2, top) and root mean square error (Figure 2, bottom). Lets define the limit of forecast skill to be the lead time for which the forecast is correlated with the observed conditions at 0.6 (has an rms error with the observed record that is equal to 2/3 of the standard deviation of Nino3.4 in the control integration). These levels are indicated by the red lines in Figure 2. What is the limit of skill for ENSO forecasts? Can you extend the limit of forecast skill above this value? Explain.

4)Now lets assume there are errors in the initial conditions. Run the model again (for 300 years) and choose option “perfect = 0” for initial conditions. How do errors in the initial conditions affect the forecast skill at 1-month? How do they affect the limit of forecast skill? Explain.

Finally, lets explore how long the instrumental record needs to be for us to get a reliable estimate of the limit of forecast skill for ENSO.

5)Re-run the model for 20, 30, 40, 50, 75, 100 and 1000 years. In each case, assume errors in the initial conditions (perfect = 0) and make forecasts for 12 months, and note the correlation at 9 months lead time and 12 months lead time, and the limit of forecast skill (i.e., where correation = 0.6). How many years of data are required to determine the true limit of forecast skill? If this model is representative of the real world, do we have enough data (30-50 years) to determine the true limit of predictability of ENSO? Do we have enough data to determine the true skill level of a forecast with a 9 month lead time? A 12 month lead time?

4c) Forecasts for a particular start time: individual forecasts

Now lets look at a single forecast using the MATLAB script “ensob.m”. This routine will create many forecasts from a single start time, using either perfect initial conditions or initial conditions that are slightly different from Truth. It explores the value of using an ensemble of forecasts. When you run the model, the output is as a function of forecast lead time and is displayed in Figure 3. In each panel, the observed Nino3.4 is shown in black. In the top panel, the observed Nino3.4 (Truth) is plotted against a single forecast, while the middle panel shows the observed Nino3.4 and has a red line for each of the Nino3.4 forecasts in the ensemble. The bottom panel shows the observed Nino3.4, the mean of the ensemble of forecasts (red solid line), and the 1-sigma error in the forecasts (red dashed lines).

6)You may have to re-run the model (type “enso”) to set up the dynamical matrix. Then type “ensob” to create 12 forecasts from the same initial conditions. You will be asked to chose the lead time and whether you want to start from perfect initial conditions or not: try fcast=6 months and perfect=0, respectively. In each plot, the black line is the same (Truth), and the red line is an individual forecast. Study these twelve plots. What do you conclude? Run the model a few more times (use the same values for fcast and perfect). What do you conclude? How predictable is ENSO?

4d) Forecasts for a particular start time: skill vs. ensemble size

Now lets look at an ensemble of forecast using the MATLAB script “ensoc.m”. This routine will create many forecasts from a single start time, using either perfect initial conditions or initial conditions that are slightly different from Truth. It explores the value of using an ensemble of forecasts. When you run the model, the output is as a function of forecast lead time and is displayed in Figure 4.

7)You may have to re-run the model (type “enso”) to set up the dynamical matrix. Then type “ensoc” to create many forecasts from the same initial conditions. You will be asked to chose the number of forecast to make, the lead time and whether you want to start from perfect initial conditions or not: try perfect=0, fcast=24 months and ncast=25, respectively. In each panel, the observed Nino3.4 is shown in black. In the top panel, the observed Nino3.4 (Truth) is plotted along with a red line for each of the Nino3.4 forecasts in the ensemble. The bottom panel shows the observed Nino3.4, the mean of the ensemble of forecasts (red solid line), and the 1-sigma error in the forecasts (red dashed lines). Study these two plots. Try a couple of more forecasts, using the same values of perfect, fcast, and ncast (each set of forecasts will be for a unique Truth). What do you conclude? Try smaller ensembles (change ncast to be 5 and 10). In light of these results and those in (6), what do you conclude? How predictable is ENSO?