MPO 674 Lecture 4

1/26/15

Lorenz (1965)

• “… weather predictions still do not enjoy the accuracy which many persons believe they have a right to expect.”

• Why?– Atmosphere not deterministic? (insignificant)– Observations are insufficient?• “tropical hurricanes may go unnoticed”

– Forecast techniques are inadequate?

Difference between tides and weather

• Tides = periodic• Weather = largely nonperiodic

• L65: if a system is varying nonperiodically, and if the present and/or past states are not known with complete accuracy, any forecasting procedure will lead to poorer and poorer forecasts as the range of prediction increases, until ultimately only the periodic component can be predicted in the far distant future.

Goal of paper

• To examine the limitations upon forecast accuracy imposed by the incompleteness and inaccuracy of observations– We now call it “initial conditions”

• What is the minimum range of prediction for which typical errors in observing the present state lead to unacceptably poor forecasts?

• Introduces a 28-variable model

Section 4

• Growth rate of errors, superposed upon a basic solution

• Errors growing during a period where they are assumed to be small, with linearized equations governing error growth holding

• Second question in Project 1A will be to compute amplification factors for L63 equations!

• Follow equations 18-35 …

64 d

32 d

16 d

8 d

4 d

2 d

Range of ‘acceptable’ forecasts

• Errors become ‘intolerable’ if they amplify by a factor of more than about 5

• Acceptable range?– Over 1 week– Less than 1 month

• Note: unrealistic wind speed, potential temperature

Section 5

• Non-linear growth• Errors grow and ultimately saturate, as the

difference between two random states• Errors in small-scale features can enter their

non-linear phase of growth quickly (e.g. convection), while errors in large-scale features can continue to grow linearly.

• Lorenz tried one quadratic non-linear function

Detailed behavior not predictable

• No single preferred course of behavior by 48 days.

• Ensemble forecast!

Some long-range predictability

• Forecast of sign of Q is predictable out to several weeks, well into non-linear phase

Conclusions

• Small random errors in observation (initial conditions) grow until they become significant features of the total field of motion

• Growth rate of errors highly variable• Small sphere of initial states becomes

deformed into an elongated ellipsoid• Important to exploit knowledge that dominant

error fields after X days can be estimated

Final remark

• “It would appear, then, that best use could be made of computation time by choosing only a small number of error fields for superposition upon a particular initial state, thus hopefully allowing the study of perturbations upon a considerable number of different initial states.”