## An example calculation using the geostrophic wind approximation.

...From USA Today

Note: In all the formulas here, / means to divide, * means to multiply, ** means the following term is an exponent(i.e. 10**(4) means 10 to the 4th power), - means to subtract, + means to add. A number followed by a "x10" to some exponent is in scientific notation to conserve space. ln( ) means to take the natural log of the variable in the parentheses. The standard rules of algebra apply to all the formulas.

The geostrophic wind approximations are broken into its two horizontal components. The "U" component represents the east-west component of the wind while the "V" component represents the north-south component. The two formulas for the U and V component of wind are as follows.

(1) U= ((Y1-Yo)(term 1)*R*Tavg(term 2)*ln(Po/P1)(term 3))/(2*omega*sin(lat)(term 4)*(D**(2)(term 5)))

(2) V= ((X1-Xo)(term 6)*R*Tavg(term 2)*ln(P1/Po)(term 7))/(2*omega*sin(lat)(term 4)*(D**(2)(term 5)))

(Y1-Yo)= the difference of the north-south component of distance between your location and the other point of reference in meters

(X1-Xo)= the difference of the east-west component of distance between your location and the other point of reference in meters

R= the average gas constant for dry air= 287 Joules/kilogram*degrees Kelvin

Tavg= the average temperature between your location and the other point of reference in degrees Kelvin

Po= atmospheric pressure at your location

P1= atmospheric pressure at the other point of reference

omega= angular velocity of rotation of earth= 7.292 x10**(-5) inverse seconds

sin(lat)= the sin function of the latitude of your location

D= distance between your location and the other point of reference in meters

These formulas are extremely tedious and complicated to use. Breaking up the equations into individual terms is probably the easiest way to tackle this calculation. Thankfully, some terms are present in both equations, which will save us some time and effort. To calculate the maximum, prevailing wind speed and direction, you will want to select two points of reference that when connected by a line, will create a line that is perpendicular to the isobars on a typical surface weather map. This will calculate the maximum pressure gradient between the two points and calculate the most likely prevailing wind.

For the example calculation, suppose you are in San Francisco and you want to estimate the wind speed between you and a weather data buoy 400 miles to the northwest, which happens to create the perpendicular line described in the previous paragraph. Let's suppose that the temperature in San Francisco is 60 degrees Fahrenheit and the temperature over the data buoy is 50 degrees Fahrenheit and let's further assume that the atmospheric pressure in San Francisco is 1004 millibars and the pressure over the data buoy is 1000 millibars. The latitude of San Francisco is approximately 37.7 degrees north. Let's start with term 1 from the equation and calculate the individual terms.

### Term 1 and 6, (Y1-Yo) and (X1-Xo)

This is where vector calculus, right triangles and sin and cosine functions come in handy. The buoy is 400 miles to the northwest, or at the end of a line that is 45 degrees to the north of west and 400 miles long. You can construct a right triangle with the 90 degree angle due west of San Francisco as in the following diagram.

You can figure out the length of sides X and Y by using trigonometric functions.

X= (The length of the hypotenuse)*cos(the angle at point C)= 400*cos(45)= 282.84 miles

Y= (The length of the hypotenuse)*sin(the angle at point C)= 400*sin(45)= 282.84 miles

Both X and Y must be converted to meters to be used in the equation. One mile equals 1610.3 meters

X (in meters)= 282.84*1610.3= 4.5546 x 10**(5) meters

Y (in meters)= 282.84*1610.3= 4.5546 x 10**(5) meters

(X1-Xo)= 4.5546 x 10**(5)

(Y1-Yo)= 4.5546 x 10**(5)

### Term 2, R*Tavg

R is already given as 287

Tavg= (60 degrees + 50 degrees)/2= 55 degrees Fahrenheit

Tavg must be converted to degrees Kelvin using Tk=5.0/9.0*(Tf-32.0) + 273.16.

Tavg(in degrees Kelvin)=5.0/9.0*(Tf-32.0) + 273.16= 285.9

R*Tavg= 287*285.9= 8.2 x 10**(4)

### Term 3, ln(Po/P1)

Po= 1004 mb, P1= 1000 mb

ln(Po/P1)= ln(1000/1004)= -3.99 x 10**(-3)

### Term 4, 2*omega*sin(lat)

omega= 7.292 x10**(-5)

sin(lat)= sin(37.7)= .6115

2*omega*sin(lat)= 8.918 x 10**(-5)

### Term 5, D**(2)

D= 400 miles

D (in meters)= 400*1610.3= 6.4412 x 10**(5) meters

D**(2)= (6.4412 x 10**(5))**(2)= 4.1489 x 10**(11)

### Term 7, ln(P1/Po)

Po= 1004 mb, P1= 1000 mb

ln(P1/Po)= ln(1004/1000)= 3.99 x 10**(-3)

### Putting it all back together

Now we are ready to plug all the terms into the original U and V equations.

(1) U= ((Y1-Yo)(term 1)*R*Tavg(term 2)*ln(Po/P1)(term 3))/(2*omega*sin(lat)(term 4)*(D**(2)(term 5)))

(2) V= ((X1-Xo)(term 6)*R*Tavg(term 2)*ln(P1/Po)(term 7))/(2*omega*sin(lat)(term 4)*(D**(2)(term 5)))

The U comp:

U= ((4.5546 x 10**(5))*(8.2 x 10**(4))*(-3.99 x 10**(-3))/((8.918 x 10**(-5))*(4.1489 x 10**(11)))= (-1.49 x 10**(8))/(3.7 x 10**(7))= -4 meters/sec.

V= ((4.5546 x 10**(5))*(8.2 x 10**(4))*(3.99 x 10**(-3))/((8.918 x 10**(-5))*(4.1489 x 10**(11)))= (1.49 x 10**(8))/(3.7 x 10**(7))= 4 meters/sec.

If you want to convert U and V from meters per second into miles per hour, use the following conversion factors.

1 hour= 3600 seconds and 1 mile=1610.3 meters

U= -4*3600/1610.3= -9 miles per hour

V= 4*3600/1610.3= 9 miles per hour

### The final steps

To calculate the final answer, we need to combine the U and V components to get the actual wind speed and direction. The easiest way is to use a vector triangle, similar to the one used previously in the distance calculations. The trigonometric functions can be utilized again as shown below.

Note: The minus sign in front of the U component answer indicates an easterly component to the wind, or in other words, a wind that blows from east to west. A positive U component would mean that the wind is blowing from west to east. The positive V component answer indicates a southerly component to the wind, or in other words, a wind that blows from south to north. A negative V component would mean that the wind is blowing from north to south.

Vector CB represents the U component of the wind

Vector BA represents the V component of the wind

Vector CA represents the actual wind vector

In order to calculate vector CA, we need to calculate the angle at point C. The tangent of an angle is the opposite side divided by the adjacent side.

In this case the tangent(angle C)=side BA/side CB= tangent(angle C)= 9/9= 1

To get angle C, take the inverse tan of 1, or angle C= tan**(-1)[1]= 45 degrees.

To get the actual velocity of the wind, you can use angle C and the sin function.

sin(angle C)= side BA/side CA

side CA(the actual wind velocity)= side BA/sin(angle C)= 9/sin(45 degrees)= 12.7 miles per hour

Since, you already know that the U component of the wind points due west, or at 270 degrees, you simply add the 45 degrees you got previous from angle C to get the final direction the wind is blowing toward. However, meteorologists want to know where the wind is coming from. To get the official direction, simply subtract 180 degrees as shown below.

Final direction= (270 + 45) - 180= 135 degrees

According to the geostrophic wind approximation and based on the conditions stated above, you would have a southeasterly wind (135 degrees) blowing at 12.7 miles per hour.