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The empirical orthogonal function, or EOF, was developed by Lorenz (1956) to develop a statistical method of weather prediction. Although this implementation of EOFs was never used widely, EOFs have been used to extract physically important modes of temporal variation from large data sets. Most of these studies have focused on finding the predominant modes for the fluctuations of fields on planetary and synoptic scales (e.g., Kutzbach, 1967; Barnett, 1983). However, Wilson (1996) and Wilson and Wyngaard (1996) have recently used EOFs to examine flow structures in the boundary layer simulated by a large-eddy simulation. In these studies, the authors found that some EOFs corresponded to gravity wave structures, while others had the form of horizontal roll vortices. In this section, we will examine what flow structures are uncovered when the EOF technique is applied to convection simulated by a mesoscale model.

The following outline of the EOF technique is similar to that of Kutzbach (1967). Let

represent a vector of observations at M different spatial locations taken at the nth time. If we are examining N times at which observations are taken, then F is an M by N matrix, with the nth column corresponding to

. We are looking for the vector

that is most similar to all of the

simultaneously. The way that Kutzbach (1967) measures similarity is by taking the normalized, squared inner product between

and F:

When the above expression is maximized, each observation vector

can be expanded as a sum of eigenvectors given by

.

Each element corresponds to the coefficient associated with the ith eigenvector for the nth observation. As Kutzbach (1967) points out, "the coefficients play the same role in See .. as the coefficients in ... a Fourier series."

We calculated EOFs for the perturbation horizontal and vertical velocity and potential temperature fields of the simulation at times ranging from 3600 to 14400 seconds. Here, perturbations are defined as deviations from the initial horizontal mean. To normalize the potential temperature perturbations and convert them to velocity units, they were multiplied by , where is the acceleration of gravity, is a Brunt-Väisälä time scale of 100 s and is the vertically-varying basic-state potential temperature. Since the calculation of EOFs is extremely memory-intensive, we had to calculate the EOFs at points that covered a limited portion of the domain (from -200 km to +200 km in the horizontal), and with limited resolution (every 4 km in the horizontal and every 1.25 km in the vertical.) With output fields saved every 120 seconds, this made for 91 "observations" at each point. A plot of the time-varying coefficients for the first five EOFs is shown in Fig. 3.

The spatial structures of the first three EOFs in u, w and ¸ are shown in Fig. 4. The first EOF (containing 12% of the variance) seems to be associated with the strengthening of the convection, due to its similarity in structure to the storm averaged perturbations, and its nearly monotonic increase in amplitude. The second and third EOFs (containing 7% of the variance apiece) are very similar (but slightly out of phase) in spatial and temporal structure. They have most of their amplitude in the stratosphere (apart from some large velocity perturbations near the region of convection), and exhibit a phase relationship indicative of outward-propagating gravity waves, as explained below. When the second and third EOFs are combined in space and time (not shown), the combined mode is similar in structure to the individual modes, except with higher amplitude.

The direction that the waves in EOFs 2 and 3 are propagating can be ascertained by examining the relative phase of vertical velocity and potential temperature. A simplified depiction of one of these waves is shown in Fig 5. On the left side of this wave, the upward motion causes the air to cool, while the downward motion on the right side of the wave is associated with warming. Since the coldest and warmest air is to the left of the peak cooling and warming, respectively, the wave propagates to the right.