CALCULATING MESOSCALE "COLDER THAN" STATISTICS
USING "MESOSCALE FLUCTUATION AMPLITUDE" MODEL

2001 March 10

Abstract

Air parcels in the real world experience temperature histories that exhibit greater fluctuations than inferred from synoptic scale trajectory data.  Differences in temperature between the actual (mesoscale) and synoptic scale version can be thought of in terms of a histogram that can be characterized by the full-width, half-amplitude parameter, also referred to as the Mesoscale Fluctuation Amplitude, MFA.  Extensive measurements of isentrope wrinkles using the Microwave Temperature Profiler have been used to create a model that relates MFA to the four independent variables:  altitude, season, latitude and underlying topography.  Thus, for any setting in which the altitude, season, latitude and underlying topography are specificed, it is possible to calcualte MFA and then determine statistics about the mesoscale versus synoptic scale temperature histogram.  This information, in turn, can be used to calculate statistics such as the "fraction of time" the real parcel is colder than the synoptic representation.  This web page illustrates the MFA concept, and describes how the "colder than" statistics can be calculated.
Introduction

The Microwave Temperature Profiler, MTP, can be used to create isentrope altitude cross-sections, IACs.  These can be used to assess how often air parcel temperature departs from a synoptic value by specified amounts.  Another web page, Mesoscale Temperature Fluctuation Model, describes in detail an analysis of data from MTP instruments aboard ER-2 and DC-8 aircraft for a variety of locations and seasons.  That web page shows that a parameter called the Mesoscale Fluctuation Amplitude, MFA, varies with altitude, season, latitude and underlying topography.  A table, and an equation, is presented that allows a user to predict MFA for any setting.  This web page describes one specific flight, ER1989.01.20, and shows by example how to predict MFA for other settings.

For the reader who wishes to see a more general overview of the evidence for the existence of mesoscale structure, click on Mesoscale Temperature Fluctuations Overview

MFA Defined

Mesoscale Fluctuation Amplitude, MFA, is defined to be the half-power full-width, HPFW, of the distribution of deviations of an isentrope surface from its synoptic scale average.  I have arbitrarily chosen to use a synoptic scale averaging procedure that gives approximately the same result as low-pass spatial frequency filtering.  My procedure is to calculate a 400-km wide boxcar average, which is subjected to a second 400-km boxcar averaging process.  The difference between the unfiltered T(t) trace and the synoptic scale version (the double-boxcar filtered version) is defined to be the mesoscale fluctuation dT((t) trace.

Sample Flight

The figure below shows the flight track of a typical ER-2 flight used in the present analysis.  It occurs under "polar winter" conditions, and is based in Stavanger, Norway.  The flight date is January 20, 1989, which is represented as 890120 in the figure titles.  Tick marks are shown at 2000 second intervals.  The portion of flight within 100 km of the coast is categorized as "coastal mountain" and it is assigned a subjective topography roughness value.  Since not all mountains have the same roughness, I have been guided by the following:

For the flight of 890120 the flight is divided into two types of flight segments, "coastal mountains" and "ocean," with topography roughness scores of 1.0 and 0.0, respectively.

Figure 1.  Flight track for ER-2 flight of 1989.01.20, with markers at 2000 second (2 ks) intervals.

Next an "isentrope altitude cross-section," or IAC, is calculated from the MTP data.  The following figure illustrates this product, showing the altitude of isentropes separated by 10 K of potential temperature.
 

Figure 2.  Isentrope Altitude Cross-section, showing the altitude of isentrope surfaces at 10 K intervals of potential temprature.

The IAC is used to choose a specific isentrope to represent flight segments.  In this example the 440 K and 460 K isentropes are used to represent the pre-dip and post-dip flight segments.  The isentrope altitudes for these flight segments is shown in the following two figures.
 

Figure 3.  Altitude of the 440 K isentrope for the first half of the 890120 flight.

Figure 4. Altitude of the 460 K isentrope for the second half of the 890120 flight.

The thick black trace in these two figures is my representation of a synoptic scale fit to the MTP data.  The departures of the red trace from the black trace are used to create a histogram of "mesoscale only" fluctuations.  Since each of the previous figures shows data for both "coastal mountains" and "ocean" categories, it was necessary to form the histograms from carefully assigned segments of the two traces.  Examples of the two histograms are shown in the following two figures.
 

Figure 5Histogram of all "coastal mountain" portions of the data shown in the previous two figures (for the 890120 flight).

Figure 6Histogram of all "ocean" portions of the data shown in the previous two figures (for the 890120 flight).

Summary of All ER-2 MFA Data

The foregoing analysis was performed for 49 ER-2 flights, comprising 73 flight segment topography assignments.  The interval of flight dates extends from 881231 to 970925.  After performing several analyses (described in more detail at Mesoscale Temperature Fluctuation Model ) a convincing case could be made for a seasonal dependence from only ER-2 data (17-21 km), shown in the next figure.

Figure 7.  Empirical model fit to winter and summer only data (including all underlying terrains), using only ER-2 data.

The thick red line is a linear fit to the "summer only" MFA data.  The dashed blue line is a fit to the winter data, and it employs a quadratic dependence upon latitude.  The equation is described in detail in the previous link.

After performing the same analysis on DC-8 MTP data, a solution was obtained for altitude dependence.  The following graph shows observed and predicted MFA (adjusted for altitude, season, latitude and underlying topography).

Figure 8.  ALL data, ER-2 and DC-8, and a fit incorporating a free parameter for the altitude dependence of MFA.

Figure 12 contains data from both the ER-2 and DC-8.  The fitted equation is:

    MFA =  (137 - 1.61 * Lat + 194 * Seas * (Lat/80)^2 + 43.6 * Topo) * (58.85 / P [mb]) ^ 0.39     (Eqn 2)
                      +/- 0.30          +/- 18                              +/- 10.4                                         +/- 0.10

where Seas and Topo are defined in the link given earlier, and P [mb] is the pressure of the isentrope surface.  Note the use of 0.39 for the pressure exponent.  The combined ER-2 and DC-8 data fit exhibits an R^2 = 0.61 and a residual MFA of 32 meters.  More data is needed to ascertain the statiscal significance of the difference between the altitude exponent fitted solution value of 0.39 and the predicted value of 0.50.  Standard errors on the estimate are shown below the 4 parameter fitted values.  In every case the SE is much smaller than the fitted value, with "signal to noise ratios" (fitted value divided by SE) = 5.4, 10.6, 4.1 and 3.9.

Specific Procedure for Simulating MFA

At this time Equation 2 is the best model representing MFA values over a wide altitude region, for all seasons, all latitudes and all topographies, and I recommend its use as a better alternative to a total disregard of the MFA effect.  There are two ways to get a specific sequence for "vertical displacement versus horizontal distance," dZ(x),  for adding to a back trajectory calculation of an isentrope surface's altitude.  The hard way is to request a copy of a program that does this, which employs an algorithm described in Section 6 at Mesoscale Temperature Fluctuations.   The easy way is to request a file of dT(t) from the author of this web page.

The following table of MFA values is based on the preceding analysis, and may be convenient for casual users wishing to estimate the possible importance of the MFA effect.  To use the table, choose a latitude region (left-most column), choose a season (center two columns), and choose an underlying terrain (right-most column), and read from the body of the table an MFA.  This MFA is what can be expected at ER-2 altitudes (19.4 km); for DC-8 altitudes, for example, multiply the MFA value by 0.61.  For other altitudes, multiply the MFA value by (58.85 [mb] / P[mb]) ^ 0.39.

TABLE 1 - MFA for ER-2 Altitudes (19.4 km)
Multiply by 0.61 for DC-8 Altitudes (11.4 km)

   (Latitude Region)     WINTER    SUMMER     (Underlying Terrain)
   POLAR      239 meters
     186    "
    68 meters
    16    "
     Mountains
     Ocean
   MID-LATITUDE      173 meters
     121    "
    125 meters
      72   "
     Mountains
     Ocean
   TROPICAL      176 meters
     124    "
    173 meters
    120    "
     Mountains
     Ocean

Exceedance Statistics

Let us ask the question:  "Given a MFA value, how often does a parcel's actual temperature depart from the synoptic scale version by specified amounts?"

The previous question will be answered for a specific flight, then it will be generalized.

Considering the coastal portion of ER1989.01.20 (altitude of 19.4 km, winter season, Arctic latitude), the MFA is 200 meters.  The following graph shows what fraction of the time the actual (mesoscale) temperature departed from the synoptic temperature by X-axis amounts.

Figure 9.  Fraction of time parcel temperature is COLDER THAN (and warmer than) the synoptic temperature by more than the X-axis amount (MFA = 200 meters, or 2 K).

Figure 10.  Fraction of time parcel temperature is COLDER THAN (and warmer than) the synoptic temperature by more than the X-axis amount (MFA = 200 meters, or 0.5 K).

Figure 9 and 10 are the same, except for one graph employing a log-scale for the Y-axis.  The green fitted line has the equation:  Y = 0.50 * (0.26X), and corresponds to MFA = 200 meters (i.e., MFA = 2.0 K).  For example, according to the equation, the mesoscale temperature will be at least 2.0 K colder than the synoptic scale temperature for 3.4% of the time.  This applies to both positive and negative departures; therefore, 3.4% of the time the parcel mesoscale temperature is at least 2.0 K warmer than the synoptic scale temperature.

For situations having other MFA values, the constant that's raised to a power is changed.  For example, when MFA is 100 meters (or 1.0 K), Y = 0.50 * (0.07X).  The general equation is:

    Y = CX
        Y = fraction of time mesoscale temperature is colder than (or warmer than) synoptic temperature by X degrees Kelvin
        where C = 0.07 (100 meters/MFA) , or C = 0.07 (1.0 K/MFA)

The following graph can be used to determine the "fraction of time colder than synoptic" parameter for various MFA values.

Figure 11.  Exceedance plots for various MFA values [meters], showing fraction of time the mesoscale temperature is colder than synoptic.  The Y-axis also corresponds to the fraction of time the mesoscale temperature is warmer than synoptic.

Consider the setting of northern Greenland (mountainous topography), winter, altitude = 26 km.  For this setting MFA is predicted to be 295 meters (or 2.95 K).  For this MFA air parcels are predicted to be colder than synoptic by at least 4.2 K about 1 % of the time.

Summary

For modelers wishing to assess the implications of mesoscale temperature fluctuations I recommend the following:

    1) request from BruceLGary@cox.net a dT(t) file that has MFA = 100 meters (1.0 K) and wind speed of 20 [m/s],
    2) calculate an MFA for the altitude, season, latitude and topography of interest,
    3) multiply the dT column of the file by the factor MFA/100 meters,
    4) multiply the time column by the ratio of wind velocities, and
    5) add the resulting table of adjusted dT(t) to a synoptic scale time series T(t) to arrive at a sample mesoscale T'(t) time series,

and run your model again with this new temperature time series to see if anything different happens (to PSC formation and evolution, for example).

I'm waiting...

Related Links

    Mesoscale Temperature Fluctuations  General introduction to the evidence for isentrope wrinkles (temperature non-uniformities) using airbore MTP data

    Mesoscale Temperature Fluctuation Model  Describing the derivation of a model for predicting MFA

    Main Menu   Meteorology and clear air turbulence web pages

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This site opened:  March 10, 2001.  Last Update:  February 27, 2002