MESOSCALE FLUCTUATION AMPLITUDE MODEL

Bruce L. Gary
Jet Propulsion Laboratory, Pasadena, CA

Abstract

The mesoscale structure of isentrope vertical displacements have been studied using ER-2 and DC-8 Microwave Temperature Profiler measurements of isentrope altitude cross-sections.  Data from 94 flight segments have been fitted to an equation with 5 free parameters, yielding a solution that allows for the calculation of Mesoscale Fluctuation Amplitude, MFA, for a range of altitudes, latitudes, seasons and underlying topography.  MFA is most strongly dependent upon an invented parameter that combines season with latitude.  The greatest MFA values are found at polar latitudes, in winter, over mountainous terrain, whereas the lowest MFA values are found during the summer season over polar latitude oceans.  MFA increases with altitude in approximate agreement with the expected "reciprocal square-root of air density" relation.  It is suggested that the MFA equation be used 1) as an additional temperature fluctuation term for back trajectory calculations, and 2) in GCM calculations that require a method for modeling the altitude region where breaking gravity waves produce momentum flux that alters wind speed.

I. Introduction

Microwave Temperature Profiler, MTP, isentrope altitude cross-sections, IACs, were used in 1987 to show that mountain waves penetrate the polar vortex tropopause and amplify with altitude in the lower stratosphere (Gary, 1989).  In 1987 and 1989 MTP also measured the existence of an ever-present background of vertical displacements of isentrope surfaces having mesoscale spatial frequences.  In polar regions, during winter, the amplitude of this structure was found to be greater for flight over land than for flight over ocean (Gary, 1989; Bacmeister and Gary, 1990).  This article presents a quantitative analysis of the amplitude of the mesoscale altitude departures of isentrope surfaces from their synoptic scale average.  The measurement of mesoscale structure amplitude is based on Microwave Temperature Profiler data from selected ER-2 and DC-8 flights that were part of missions sponsored by NASA's Upper Atmosphere Research Program during the period 1989 to 2000.

A companion article, "Mesoscale Temperature Fluctuations," presents the case for the existence of mesoscale structure using airborne data.  MTP IACs are shown to exhibit more structure than IACs based on synoptic scale "analyzed" temperature and wind fields.  In the present analysis, mesoscale structure is defined to be the difference between the measured structure and a 400-km double boxcar average (described below).  Since air parcels traveling through isentrope structures will undergo temperature fluctuations, the structure is also referred to as mesoscale temperature fluctuations.  The amplitude of these vertical displacements can be quantified, using histograms, and in this article the Mesoscale Fluctuation Amplitude parameter is used to describe histogram widths.

This article is relevant for two types of investigations: 1) back trajectory studies where mesoscale vertical motions and their associated temperature fluctuations are important properties of air parcel histories, and 2) climate modeling of horizontal momentum flux caused by the breakdown of gravity waves that have been amplified at high altitudes.  It is important to assign realistic values of momentum flux to the wind field at the proper altitudes, and one of the present shortcomings of modeling studies is the lack of an adequate model for representing the magnitude of vertical displacements of gravity waves at altitudes below those where the wave breakdown occurs (Bacmeister et al, 1994).

II. Data Analysis Procedure

Mesoscale Fluctuation Amplitude (MFA) is defined to be the half-power full-width, HPFW, of the distribution of altitude departures of an isentrope surface from a synoptic average.  This study employs a synoptic averaging procedure that gives approximately the same result as low-pass spatial frequency filtering.  An isentrope's altitude is averaged using a sliding 400-km uniform weighting function (a 400-km wide boxcar average), which is again subjected to a second 400-km boxcar averaging.  Histograms of departrues from this synoptic scale representation are then fitted to a Gaussian shape with manual adjustments of offset, height and width, using a spreadsheet application.  The resulting MFA determinations are entered into a data base, along with several independent variables:  latitude, date, an underlying topography roughness parameter and average altitude of the isentrope.  Multiple regression analyses are eventually performed on the entire data base using various combinations of independent variables, with MFA as the dependent variable.

The following several figures are used to illustrate a typical analysis of a flight and the determination of MFA entries into a data base.

Figure 1 is the flight track of a typical ER-2 flight used in the present analysis.  It occurs under "polar winter" conditions, and was based in Stavanger, Norway.  The flight date is January 20, 1989, also referred to as flight ER1990.01.20.  Tick marks are shown at 2000 second intervals of Universal TIme (UT).  The following list was used to guide the subjective choice for an underlying topography roughness parameter, referred to below as "topography":

         Land Type                    Topography Parameter, "Topo"

    Ocean                0.0
    Flat Land            0.4
    Coast                0.6
    Coastal Mtns         0.5 - 0.8
    Continental Mtns     0.6 - 1.0

Flight ER1989.01.20 consists of two types of flight segments, "coastal mountains" and "ocean," with "topo" scores of 0.8 and 0.0, respectively.  Data is assigned to "coastal mountains" when the ER-2 was within 100 km of the coast.

An "isentrope altitude cross-section," or IAC, is calculated from the MTP data.  Figure 2 illustrates this product, showing the altitude of isentropes separated by 10 K of potential temperature.  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 first and second halves of the flight.  The isentrope altitudes for the the first half flight segment is shown in Figure 3.

The thick black trace in this figure is a synoptic scale fit to the MTP data, derived using the double 400-km boxcar procedure.  The departures of the thin trace from the thick trace are used to create a histogram of "mesoscale only" fluctuations.  Since this data includes flight over both "coastal mountains" and "ocean" categories, it was necessary to create separate histograms from carefully assigned flight segments. Examples of the "coastal mountain" and "ocean"  histograms are shown in Figures 4 and 5.

III. ER-2 Data

At the present time the data base for ER-2 flight data is based on 49 ER-2 flights from the interval 1988.12.31 to 2001.09.26, yielding 73 MFA values corresponding to a range of latitudes, seasons and topographies.  All isentrope altitudes are within the range 17 to 21 km, with an average altitude of 19.4 km.  Figure 6 shows all 73 MFA values plotted versus latitude.  The general pattern in this figure is that "MFA variability increases with latitude."   By considering only "winter" and "summer" data the underlying cause for the increased variability with latitude can be deduced.  Figure 7 presents the same data with summer and winter represented by different symbols (an "outlier" associated with a jet stream is not used).  MFA is largest at high latitude winter and lowest at high latitude summer.  The straight lines correspond to the following equations:

    MFA [meters] = 120 -  0.7 * Latitude [degrees], for summer data                                                (Eqn. 1)
    MFA [meters] = 120 + 1.0 * Latitude [degrees], for winter data                                                   (Eqn. 2)

Whereas there appears to be negligible seasonal variation in the tropics, the amplitude of seasonal variation appears to increase linearly with latitude.  In order to sort the data according to season it was necessary to invent a "season parameter" based upon the flight date's month and day of month.  The parameter "Winterness" was defined according to the following equation:
 

        Winterness = (1 + sin (  * (MV - 10.7)/6))/2                                                                             (Eqn. 3)

where MV = month number + (day of month) / 31, and month number =1 for January, etc.  Winterness varies smoothly in a sinusoidal manner from 0.0 on July 22 (one month after summer solstice) to a value of 1.0 on January 22 (one month after winter solstice).  In Fig. 7 MFA data are categorized as "summer" or "winter" on the basis of Winterness being less than or greater than 0.5.

Figure 7 demonstrates that MFA at ER-2 altitudes depends upon latitude and season.  A possible dependence upon underlying topography was investigated by performing a 3-term least squares (LS) fit of measured MFA versus the following three independent variable: 1) topography parameter, 2) Latitude, and 3) Winterness * Latitude.  The following solution was obtained:

     MFA' = 116 +188 * Topography -1.17 * Latitude +2.00 * Winterness*Latitude                       (Eqn. 4)
                         +/-85                    +/-0.23               +/-0.17

where MFA' [meters] is a model predicted MFA, "Topography" is the underlying topography parameter (ranging from 0 to 1), "Latitude" is latitude [degrees] and "Winterness"  is [dimensionless].  Formal standard errors of the coefficient solutions are shown below each coefficient. This fit exhibits r2 = 0.68, and has a residual MFA of 30.5 meters.  Figure 8 shows the relation between measured MFA (from ER-2 flights only) and predicted MFA' using the above equation.

The four constants in the above equation have values significantly different from zero; the ratios for "parameter value to parameter value uncertainty" are 2.17, 5.08 and 11.67.  (When small adjustments are made for an altitude dependence, described below, this 3-parameter solution is "stronger," yielding a topography "parameter value to uncertainty ratio" of 2.45)  All independent variables are statistically significant.  From this analysis of ER-2 data it can be concluded that:

    1) there is a strong latitude dependence,
    2) there is a strong seasonal dependence, and
    3) there is a moderately significant topography dependence.

IV.  DC-8 Data and the Altitude Dependence of MFA

This section shows that MFA depends upon altitude as well as latitude, season and underlying topography.  Since the foregoing analysis was with only ER-2 cruise flight data, and is confined to a rather narrow altitude region (17 to 21 km), it should be possible to investigate the altitude dependence of MFA by comparing the ER-2 MFA values with MFA results from the lower-flying DC-8 aircraft. Gravity wave theory predicts that mountain wave amplitude should increase with altitude in accordance with the relation:

      Ai = A0 * (Di/D0)-0.5                                                                                                                   (Eqn. 5)

where A0 is an amplitude constant, and D0 and Di are air density at a standard altitude and an altitude of interest.  Since air temperature is approximately constant throughout the altitude region between typical DC-8 and ER-2 flight altitudes, air density will be approximately proportional to air pressure.  This predicted dependence of wave amplitude versus altitude within the stratosphere was observed by the ER-2 during encounters with mountain waves over Antarctica (Gary, 1989).  What is true of mountain waves may not be true of the ever-present background of gravity waves, so it is necessary to verify the expected dependence of MFA on altitude.  To the extent that air density is proportional to air pressure, which it will be when air temperature is uniform between the DC-8 and ER-2 altitudes, air pressure at flight altitude can be used to represent air density.  The following analysis uses air pressure as an independent variable.

Two"outliers" in the DC-8 MFA data base were removed from consideration due to their association with large amplitude mountain waves (over the Rocky Mountains) and a sub-tropical jet stream (over the Pacific Ocean).  These two DC-8 outliers, and the outlier ER-2 MFA datum that was described above, suggest that the MFA models developed in this analysis cannot be used in situations characterized by jet streams or mountain waves.

To illustrate the necessity of invoking an altitude dependence of MFA, the combined ER-2 and DC-8 data sets are plotted against a model that does NOT include an altitude correction term, shown as Fig. 9.  The model predictions of MFA are clearly too high for the DC-8 data.  When a "reciprocal of square-root of pressure" correction is incorporated in the model, in accordance with mountain wave theory, an acceptable plot of measured versus model predicted MFA is produced, and shown as Fig. 10.  A best fit value for the pressure exponent is approximately -0.43, with an SE uncertainty that is estimated to be 0.10 (a formal uncertainty could not be obtained for this parameter using the fitting procedure of this analysis).  The MFA measurements are compatible with the expected altitude exponent of -0.50.

The final equation for representing the combined ER-2 and DC-8 sets of measured MFA involves four independent variables: 1) latitude, 2) a season/latitude parameter, 3) topography, and 4) an "altitude exponent" for use with the ratio of air pressure to a reference air pressure (58.85 millibars, the ER-2 average).  The fitting procedure was to adopt various values for the altitude exponent and then perform a standard LS fit for the remaining independent variables.  The following equation was obtained for relating MFA to the four independent variables:

    MFA =  (112 - 1.21 * Latitude + 2.20 * Winterness * Latitude + 29.0 * Topography) * (P[mb]/58.85)-0.43                    (Eqn 6)
                        +/-0.27               +/-0.20                                     +/-10.2                                              +/-0.10 (est.)

where MFA has units of meters, P is air pressure [mb], and the other parameters are described above.  This equation provides a fit to the combined ER-2 and DC-8 data that exhibits r2 = 0.645 and a residual MFA of 36.6 meters.

In every case the uncertainty of the solution's coefficient is significantly smaller than the coefficient value, with "value to SE uncertainty" ratios = 4.55, 11.27 and 2.84 (the 4th solved-for parameter is the exponent which has only an estimated uncertainty).  The fact that this last ratio increased from its "ER-2" only counterpart (i.e., increased from 2.45 to 2.84) shows that the correlation of measured MFA with underlying topography was strengthened by including the DC-8 data.  More data is needed to ascertain the statistical significance of the difference between the altitude exponent fitted solution value of -0.43 +/- 0.10 (estimated SE) and the predicted value of -0.50.

V. Suggested Procedure for Simulating MFA

At this time Eqn. 6 is the best model representing MFA values throughout a range of altitudes, seasons, latitudes and topographies.  Using Eqn. 6 is probably a better alternative than a total disregard of the MFA effect for investigations in which temperature fluctuations are potentially important.  Eqn. 6 also provides a means for incorporating realistic wave amplitudes in a synoptic field intended for use in calculating gravity wave breakdown at high stratospheric altitudes.  It has the virtue of not requiring a detailed model for generating the waves and may therefore be suitable for operational use.

There are two ways to calculate a specific sequence for "vertical displacement versus horizontal distance," dZ(x) to be added to specific back trajectory calculations of an isentrope surface's altitude.  The difficult way is to request a copy of a program that does this, which employs an empirical algorithm developed for this purpose.  The easy way is to request a copy of a file "dZ(x)" from the author (BruceLGary@cox.net).  The user may then modify the dZ column by a multiplication factor to convert it from having the standard MFA value of 100 meters to the desired MFA.  If a specific dZ(x) function is not required, but a probability density distribution for dZ is adequate, then this can be calculated from the following simple equation:

     P(dZ) = EXP((-dZ/0.60*MFA)2)                                                                                                (Eqn. 7)

where "EXP" means "e raised to the power" and dZ is altitude departure from a synoptic average.  This equation is normalized such that P(0) = 1.  MFA must be calculated from Eqn. 6, above, before using this equation.

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 a MFA value from the body of the table. This is a "most likely" MFA for ER-2 altitudes (19.4 km); for DC-8 altitudes, for example, multiply the MFA value by 0.58.  For other altitudes, multiply the MFA value by (58.85 [mb] / P[mb])0.43.

                    TABLE 1 - MFA for ER-2 Altitudes (19.4 km)
                              Multiply by 0.58 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

VI. Speculations on the Causes for Observed Correlations

The fact that MFA is greater for "winter high latitudes" suggests that the independent variable created for the multiple regression analysis is merely a "proxy parameter" for wind speed.  Support for this comes from the fact that two of the high-value MFA outliers were for flight along jet streams.  The fact that clear air turbulence (CAT) occurs preferentially during the winter season, which is conventionally attributed to the greater wind speeds in winter, is also supportive of this interpretation - since CAT occurs when Kelvin-Helmholtz waves are amplified by strong vertical wind shears in the presence of insufficient static stability (provided by the temperature field).

The dependence of MFA upon rough underlying terrain seems to require that the source for MFA is the uplifting of air by terrain irregularities, causing vertical displacements that grow with altitude.  This is supported by the near-absence of a correlation of MFA with underlying terrain during the polar summer, when winds are light.

VII. Conclusion

Modelers wishing to assess the magnitude of mesoscale temperature fluctuations may use Equation 6 (or Table 1) to calculate MFA.   The MFA value is needed to scale sample sequences of dZ(x), available from the author, which can then be added to a synoptic scale version of an air parcel's altitude versus back trajectory distance.  Modelers wishing to calculate the altitude where wave breakdown occurs may use this same MFA as a most likely value when it is not feasible to employ an explicit calculation of mountain wave amplitude.

The fact that MFA depends upon seaon, latitude, underlying topography and altitude should provide useful clues for guiding theoretical investigations into the origin of the atmospheric waves producing these ever-present mesoscale fluctuations.

References

Bacmeister, J. T. and B. Gary, "ER-2 Mountain Wave Encounter Over Antarctica: Evidence for Blocking," Geophys. Res. Lett., 17, 81-84, 1990.

Bacmeister, J. T., P. A. Newman, B. L. Gary, K. R. Chan, "An Algorithm for Forecasting Wave-Related Turbulence in the Stratosphere," Weather and Forecasting, 9, 2, June 1994.

Gary, Bruce L., 1989, "Observational Results Using the Microwave Temperature Profiler During the Airborne Antarctic Ozone Experiment," J. Geophys. Res., 94, 112223-11231.
 

Figure Captions

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

Figure 2.  Isentrope Altitude Cross-section, showing the altitude of isentrope surfaces at 10 K intervals of potential temprature for the ER1989.01.20 flight.

Figure 3.  Altitude of the 440 K isentrope for the first half of the ER1989.01.20 flight.

Figure 4.  Histogram of all "coastal mountain" portions of the ER1989.01.20 flight.

Figure 5.  Histogram of all "ocean" portions of the ER1989.01.20 flight.

Figure 6.  Plot of ER-2 MFA values versus latitude.  All seasons and topography types are included.

Figure 7.  "Winter" and "summer" MFA data versus latitude, with arbitrarily placed straight lines to indicate the seasonal differences between MFA versus latitude.  All data are from ER-2 flights.

Figure 8.  Measured MFA versus MFA predicted by an empirical model least squares fit making use of the three independent variables: latitude, a latitude/sesonal term (the product of "winterness" with latitude), and underlying topography.  All data are from ER-2 flights.

Figure 9.  Measured ER-2 and DC-8 MFA data are plotted versus MFA predictions based upon the ER-2 MFA analysis with no provision for an altitude dependence of MFA.

Figure 10.  Measured ER-2 and DC-8 MFA data are plotted versus MFA predictions using a model that employs an altitude dependence of MFA specified by a pressure exponent of -0.5.

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