**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) / VWS^{2}

where g = the acceleration
of gravity, 9.74 [m/s^{2}] 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) * VWS^{2} / (dTH/dZg)

When the following units are used: TH [K], g [m/s^{2}],
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) * VWS^{2} / (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.

**GRAVITY ACCELERATION VERSUS ALTITUDE AND LATITUDE**

Pressure Altitude | Equator | 30 Degrees Latitude | 60 Degrees Latitude | 70 Degrees Latitude |

20 km | 9.719 [m/s^{2}] |
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 |

**ADIABATIC LAPSE RATE VERUS ALTITUDE AND LATITUDE**

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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*This site opened: August
1, 2000. Last
Update: Septeber 28, 2000*