Simple equations for conversion of vector wind components, speed and direction.

In this document it is assumed that all trig functions use or return angles in radians. However, angles are converted to degrees to simplify the discussion.

Radians are converted to degrees by multiplying by DperR (180 / π = 57.29578), and degrees are converted to radians by multiplying by RperD (π / 180 = 0.01745329).

This discussion assumes that the two-argument arctangent function, `atan2(y,x),` returns the arctangent of `y/x` in the range -π to π radians, -180 to 180 degrees. C, C++, Python, Fortran, Java, IDL, MATLAB and R all follow this convention.

Warning: Spreadsheets, including Microsoft Excel, LibreOffice Calc and Google Docs switch the arguments, so that `atan2(x,y)` is the arctangent of `y/x`.

To check your software, compute `atan2(1,-1)`. If it equals 2.36 radians (135 degrees) then your software uses the programming language convention and you can use these formulas unchanged. If it equals -0.79 radians (-45 degrees) then your software follows the spreadsheet convention and you must switch the arguments of atan2 in the following equations.

A positive U_{geo} component represents wind blowing **to** the East (confusingly known as a "westerly"). +V_{geo} is wind to the North (a "southerly" ). This is right handed with respect to an upward +Wgeo.

Dir_{geo} is the direction with respect to true north, (0=north,90=east,180=south,270=west) that the wind is coming **from**.

Dir_{geo}= atan2(-U_{geo},-V_{geo}) * DperR = 270 - (atan2(V_{geo},U_{geo}) * DperR)

Spd is the horizontal wind speed:

Spd = sqrt(U_{geo}^{2}+ V_{geo}^{2})

To convert from Dir_{geo} and Spd to U_{geo} and V_{geo}:

U_{geo}= -Spd * sin(Dir_{geo}* RperD) V_{geo}= -Spd * cos(Dir_{geo}* RperD)

Sonic anemometers measure wind vector components which are relative to the orientation of the sonic array. Therefore, the orientation of the array must be determined, usually with a theodolite or compass, before the wind vectors can be rotated into geographic coordinates. The following discussion assumes that the sonic anemomenter is level, so that W_{inst}= W_{geo}.

+U_{inst} represents wind into the array, parallel to the support boom (usually toward the tower), i.e. wind from the un-obstructed direction. If you're looking into the array, along the boom, toward the tower, then +V_{inst} is to your left. These are right handed with an upward +W_{inst}.

CSAT3 coordinates are like ATIs. +U_{inst} represents wind into the array from the un-obstructed direction, parallel to the support boom. U,V and W are right handed.

The R2 V_{inst} direction is 120 deg counter-clockwise from N arrow on top of the array (or the front support arm). V_{inst} is 90 degrees counter-clockwise from the pointing direction of the upper, front transducer arm. U_{inst} and V_{inst} are left handed with an upward W, so software must flip the sign of one of them to get a right handed system. If one flips V_{insti}, then these coordinates are 30 deg c-c-wise from ATI coords (if N arrow is aligned parallel to ATI boom), but 180 deg from Gill R3 coords.

The R3 U_{inst} direction is antiparallel to the unflipped U_{inst} on the R2, and so UVW are right handed. The +V_{inst} direction is 120 deg c-c-wise from the N arrow on top of the array. The R3 V_{inst} direction is 90 degrees c-c-wise from the pointing direction of the upper, front transducer.

Determine the angle with respect to true north, (0=N, 90=E) of the +V_{inst} direction. Call this angle `V _{az}`. Looking from above, the instrument coordinate system is therefore rotated

Dir_{inst}= atan2(-U_{inst},-V_{inst}) * DperR Dir_{geo}= Dir_{inst}+ V_{az}

For ATI and Campbell CSAT3 sonics, `V _{az}` is the direction relative to true north, straight into the array from the un-obstructed direction, minus 90 degrees.

For Gill R2 sonics, if the sign of V is flipped, then `V _{az}` is the angle of the N arrow + 60. If the sign of U is flipped, then

For Gill R3s, `V _{az}` is the N arrow direction + 240 degrees.

U_{geo}= U_{inst}* cos(V_{az}*RperD) + V_{inst}* sin(V_{az}*RperD) Vgeo = -U_{inst}* sin(V_{az}*RperD) + V_{inst}* cos(V_{az}*RperD)

U_{inst}= U_{geo}* cos(V_{az}*RperD) - V_{geo}* sin(V_{az}*RperD) V_{inst}= U_{geo}* sin(V_{az}*RperD) + V_{geo}* cos(V_{az}*RperD)

The U axis (U_{stream}) of streamwise coordinates is defined to be the mean wind direction. To rotate wind vectors to streamwise coordinates, first determine the the average wind vector,`U _{av}, V_{av}`, in the same coordinate system as the data to be rotated, which could be instrument or geographic coordinates. The rotation angle is the angle of this wind vector from the U axis, measured positive counter-clockwise.

D = atan2(V_{av},U_{av}) * DperR U_{stream}= U * cos(D*RperD) + V * sin(D*RperD) V_{stream}= -U * sin(D*RperD) + V * cos(D*RperD)

As expected, if `U=U _{av}` and

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