The following is a derivation of Reciprocal Richardson Number, RRi.

Derivation of Reciprocal Richardson Number, RRi

Richardson Number, Ri, is loosely defined as the ratio of "stabilizing forces" to "overturning forces."  Stabilizing occurs when buoyancy forces restore a vertically displaced air parcel to it's time-average "resting" altitude.  Thus, an atmosphere in which temperature increases by large amounts with altitude is more "stable" than one that has an adiabatic or sub-adiabatic temperature vertical gradient.  Overturning forces derive from the wind field, and have the potential to exist when horizontal wind speed, or direction, varies with altitude.  A slight vertical displacement of an air parcel will cause it to momentarily become entrained by air in the "visited" layer and this will start it moving in a different direction, or at a different speed, compared with the air at the air parcel's "resting" altitude.  The formula for Ri is usually given in the following form:

    Ri = (g/TH) * (dTH/dZg) / VWS2

        where g = the acceleration of gravity, 9.74 [m/s2] at ER-2 altitudes and mid-latitudes (see table below),
                  TH = theta, potential temperature, related to physical temperature and air pressure by TH = T * (1000 [mb] / P [mb]) 0.286
                               where T = physical temperature [K] and P = air pressure [mb]
                  Zg = geometric altitude,
                  VWSg = vertical wind shear, the scalar value of the vertical gradient of the vector difference of horizontal wind vector,
                               and VWS = ((dU/dZg)2 + (dV/dZg)2)1/2,
                                       where U and V are the two horizontal components of wind.

It is to be noted that the vertical gradients in the Ri formula are with respect to geometric altitude, not pressure altitude.  Aircraft routinely measure pressure altitude, and gradients with respect to pressure altitude can be easily converted to gradients with repsect to geometric altitude using the simple formula:

    dZp/dZg = Tstd/T, where Tstd is the "standard atmosphere" temperature for the pressure altitude of flight.

I have developed a method for deriving VWS from in situ temperature, pressure and horizontal wind measurements (described in Patent #3), which is described below:

     1) Calculate dU/dt, dV/dt and dTH/dt (where TH = theta, potential temperature), using a 10-second time difference (can also be 20-seconds, 30-seconds, etc),
            (As an alternative, perform a muiltiple regression of U and V against both time and TH, and use only the TH-derivatives; skip to step 5, below.)
     2) Delete entries with dTH/dt = 0,
     3) Calculate dU/dTH = dU/dt divided by dTH/dt, and the same for dV/dTH,
     4) Calculate weighted-average of dU/dTH and dV/dTH, using the absolute value of dTH/dt as weights (using entry & 4 or more neighbor entries),
     5) Calculate orthogonal sum of dU/dTH and dV/dTH, and call this dUV/dTH (the scalar value of the gradient of wind vector with theta),
     6) Determine dZp/dZg = Tstd/T (where, for example, Tstd = 216.65 [K] between 11 and 20 km pressure altitude),
     7) Obtain measured dT/dZp from the Microwave Temperature Profiler,
     8) Calculate dT/dZg = dT/dZp * dZp/dZg,
     9) Calculate dTH/dZg = (dT/dZg + 9.74 [K/km]) * (1000 / P [mb])0.286, where 9.74 [K/km] is the adiabatic lapse rate for typical ER-2 conditions (see table below),
    10) Calculate VWS = dUV/dTH * dTH/dZg,
    11) Average VWS using a 10-second boxcar.

Once VWS has been calculated, Ri is easily derived using the first equation, above, and the value for dTH/dZg from step 9.  Ri is calculated, and immediately converted to RRi.  RRi is then subjected to time-averaging for various averaging intervals, typically 1 or 2 minutes in length (and of the boxcar type).  RRi is a better parameter for time-averaging because the main source of measurement error in the preceding algorithm is VWS (due to the presence of horizontal wind gradients that can map-over to vertical gradients when a time series is used to infer the vertical gradient), and it is therefore better to have VWS in the numerator than the denominator when averaging is to be performed.

Since the preferred parameter, for averaging purposes) is RRi, we have:

    RRi = (TH/g) * VWS2 / (dTH/dZg)

When the following units are used:  TH [K], g [m/s2], U and V [m/s], VWS [m/s per km], dTH/dZg [theta K/km], the formula for RRi becomes:

    RRi = 0.001 * (TH/g) * VWS2 / (dTH/dZg)

Typically, RRi is averaged over a 30 to 60 second period before being subjected to warning thresholds.

In any actual embodiment of this algorithm the averaging will have to be for only data that has already been acquired, meaning that boxcar averaging centered on the time in question cannot be performed.  For such an embodiment an exponential fall-off averaging is suggested, such that the most recent measurement is added to the previous average value according to the formula

    RRi Exponential Average = 0.99 * (Previous RRi Exponential Average) +0.01 * (Latest Estimate of RRi)

which produces a trace versus time, RRi(t), similar in shape to the 1-minute boxcar average trace but offset about 30 seconds, such that it is "behind" the boxcar trace.  Thus, in a real-time system the warning times will be shortened by about 1/2 minute from those presented here using a boxcar averaging.  I prefer to use the boxcar average algorithm for this study as it should afford the best chance of gaining insight into what is actually going on, dynamically, with the air the ER-2 flew through.  This, by itself, is a worthy goal.


 Pressure Altitude    Equator     30 Degrees Latitude    60 Degrees Latitude     70 Degrees Latitude
    20 km       9.719 [m/s2]       9.732        9.758       9.764
    10 km       9.780       9.762       9.788       9.795
    Surface        9.780       9.793       9.819        9.826


  Pressure Altitude        Equator    30 Degrees Latitude    60 Degrees Latitude    70 Degrees Latitude
    10 km       -9.68 [K/km]       -9.69        -9.72       -9.73
    10 km       -9.71       -9.72       -9.75       -9.76
    Surface       -9.74       -9.75       -9.78       -9.79


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