METEO 300
Fundamentals of Atmospheric Science

8.5 Gradients: How to Find Them

The gradient of a variable is just the change in that variable as a function of distance. For instance, the temperature gradient is just the temperature change divided by the distance over which it is changing: ΔT/ΔdistanceThis equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers. . The gradient is a vector and has a direction as well as magnitude.

Surface temperature map for North America
Surface temperature map for North America on 8 September 2012. Temperatures are in Fo. For point in western Kentucky, the temperature gradient is to the southeast.
Credit: Unisys

Consider the surface temperature contour plot from NOAA for 8 September 2012 in the figure above. A strong temperature variation is draped across the eastern and southern US from New York down to Texas. How do we quantify this temperature variation? First we have to specify where we want to measure the temperature gradient. Then we simply need to choose isotherms on either side of the point, take the difference between the isotherms, figure out how far apart they are in horizontal distance, and divide the temperature change between the isotherms by the distance between the isotherms. The direction for the gradient is on the normal (perpendicular to the isotherms) from the lower temperatures to the higher temperatures. It’s pretty easy to figure out where the gradient vector points just by quick examination, but it is a little harder to figure out what the gradient magnitude and actual direction are.

Now watch this video (2:12) on finding distances:

Click here for a transcript of Finding Distances Video

Mathematically, if we know the algebraic expression for the temperature change, such that T = T(x,y), we can find the gradient by using the del operator, which is also called the gradient operator.

Recall the del operator:

  = i x + j y + k z This equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers.

If we are looking only at changes in x and y, then we can define a horizontal del operator:

H  = i   x +  j  y
[8.10]

At any point, we can determine the gradient of the temperature:

H T = i   T x +  j   T y
[8.11]

Note that this quantity has dimensions of θ/LThis equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers. and a magnitude and a direction. The gradient direction is always normal to the isolines and pointing in the direction of an increase. We can define the normal vector, which is just the unit vector in the direction of the increasing temperature. We will call this normal vector n.

See text above
Example of a gradient and the math required to calculate the gradient magnitude and direction.
Credit: H.N. Shirer

We can calculate a gradient for every point on the map, but to do this we need to know the change in the temperature over a distance that is centered on our chosen point. One approach is to calculate the gradients in the x and y directions independently and then determine the magnitude by:

| H T |= ( T x ) 2 + ( T y ) 2 = | T n |
[8.12]

and the direction by:

μ=  tan 1 ( T y T x ) This equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers.
[8.13]

We can program a computer to do these calculations.

However, often we just want to estimate the gradient. The gradient can be determined by looking at the contours on either side of the point and computing the change in temperature over the distance. These partial derivatives can be approximated by small finite changes in temperatures and distances, so that This equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers. is replaced by ΔThis equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers. in all places in these equations. We can calculate gradients by using “centered differences” as shown in the figures below.

See text aboveSee the text above
Calculating the temperature gradient in the x (top) and y (bottom) directions using the centered difference method.
Credit: H.N. Shirer

We then calculate the magnitude with Equation [8.12] and the direction with Equation [8.13], where we replace the partial derivatives with the small finite differences in all places in these equations.

See the text above
Example of an estimation of the gradients in the x and y directions. To get an idea of the horizontal scale, you can estimate distances using the known size of a state or country, in this case, Pennsylvania, which is on average 470 km (254 nm, nautical miles, 290 miles) in the x (east-west) direction and 250 km (135 nm, 155 miles) in the y (north-south) direction for the parts where the north and south borders are parallel lines.
Credit: H.N. Shirer

The magnitude and direction are:

| H T |= ( ΔT Δx ) 2 + ( ΔT Δy ) 2 =  ( 4 F o 45nm ) 2 + ( 4 F o 84nm ) 2 =0.1 F o /nm μ=  tan 1 ( ΔT Δy ΔT Δx )= tan 1 ( 4 84 4 45 )= 28 o , which points to the southeast

When you calculate the arctangent, keep in mind that the tangent function has the same values every 180o, or every π in radians. If you get an answer for the arctangent that is 45o, how do you know whether the angle is really 45o or 45o+180o=225o? The gradient vector always points toward the higher temperature air, so always choose the angle so that the gradient vector points toward the warmer air.

Recap of the process for calculating the temperature gradient:

  1. Determine the distance scale by any means that you can. Sometimes it is given to you; sometimes you can scale off a ruler; sometimes you just estimate it using the size of known boundaries.
  2. Determine the spacing between the isotherms.
  3. Find the temperature change in the x and y directions using the centered difference method. These two numbers - ΔT/Δx and ΔT/ΔyThis equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers. - are needed to calculate both the gradient magnitude and the gradient direction. Note that ΔT/Δx and ΔT/ΔyThis equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers. can be either positive or negative.
  4. Calculate the magnitude by finding the square root of the squares of the gradients in the x and y directions (i.e., ΔT/Δx and ΔT/ΔyThis equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers. ).
  5. Calculate the direction of the gradient vector by finding the arctangent of the y-gradient divided by the x-gradient. Pay attention to the direction - make sure that it points toward the warmer air.

Now watch this video (3:52) on finding gradients:

Click here for a transcript of Finding Gradients Video

A word about finding gradients in the real world. Sometimes the centered differences method is difficult to apply because the gradient is too much east-west or north-south. For instance, in the temperature map at the beginning of this section, the x-gradient is hard to determine by the centered difference method in the Oklahoma panhandle and the y-gradient is hard to determine in central Pennsylvania because in both cases, the temperature hardly changes. In these cases, you could say that the gradient in that direction equals 0, but then your computer program might have a hard time finding the arctangent. One way around this problem is to put in a very small number for the gradient in that direction, say 1 millionth of your typical gradient numbers, to do the calculation.

A second word about finding gradients in the real word. When you are finding temperature gradients from a temperature map, it is sometimes hard to determine the temperature gradient at some locations because the isotherms are not evenly spaced and can be curvy. Don't despair! Use your best judgment as to what the gradients are, Check your answers for the magnitude and direction of the temperature gradient vector by estimating the magnitude and direction by eyeballing the normal to the isotherms at that location and pointing the gradient vector to the warmer air. If your calculated direction is 160 degrees when your eyeball check says about 220 degrees, check your math again.

Quiz 8-3: Grading your gradients.

  1. Find Practice Quiz 8-3. You may complete this practice quiz as many times as you want. It is not graded, but it allows you to check your level of preparedness before taking the graded quiz.
  2. When you feel you are ready, take Quiz 8-3. You will be allowed to take this quiz only once. Good luck!