9.2: Main Forces
- Page ID
- 46259
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)There are five forces that influence the speed or direction of horizontal winds.
- Pressure gradient force
- Advection
- Centrifugal force
- Coriolis force
- Turbulent drag
Remember from Chapter 1 that according to the Cartesian coordinates, \(x\) points east, \(y\) points north, and \(z\) points upwards. To define wind, we use wind components \(u, v\), and \(w\) which correspond to the \(x, y\), and \(z\) directions. These wind components are used in equations of motion used to predict wind to designate direction in a three-dimensional plane. We’ll go through each of the five forces one by one to discuss how they affect wind speed and/or direction.
Pressure Gradient Force
A pressure gradient (\(PG\)) is a change in pressure over a distance. Therefore, the units of the pressure gradient force are Pascals per meter (P m-1). The pressure gradient can be calculated simply as the change in pressure divided by the distance over which that change occurs. The size or strength of the pressure gradient determines the size or strength of the force that results from it.
\[P G=\frac{\Delta P}{\Delta x} \nonumber \]
The pressure gradient force (\(PGF\)) is a force from high to low pressure over a distance. Without differences in pressure, there would be no wind because there would be nothing to accelerate airflow. In Chapter 9 we learned about isobars, or lines of constant pressure. When isobars are tightly packed, we know that there is a strong pressure gradient or large change in pressure over a relatively short distance. A strong pressure gradient results in a large pressure gradient force and, as we’ll see, higher wind speeds. Depending on the text you read, you’ll see a number of different forms of the pressure gradient force. Two equivalent representations are given below.
\[\frac{PGF_x}{m}=-\frac{1}{\rho} \cdot \frac{\Delta P}{\Delta x} \nonumber \]
The \(PGF\) as defined above is simply a change in pressure divided by the distance and air density. This gives the force per unit mass, hence \(PGF/m\) . Above we define \(PGF\) in the \(x\)-direction, but it can act in any horizontal plane. The units of the \(PGF\) will be in units of acceleration above because Force = Mass * Acceleration, but because we’re dividing the left-hand side by Mass the units will be m s-2. The negative signs in the above equation is due to the fact that the PGF acts from high pressure to low pressure.
Another equivalent way to represent the \(PGF\) is as follows.
\[|P G F|=\text { Volume } \cdot \frac{\Delta P}{L} \nonumber \]
Because Mass = Volume * Density, we then see that Volume = Mass/Density. The second equation is identical to the first, except that Mass/Density is on the right hand side in the form of Volume instead of being split between left and right sides of the equals sign. Also, the variable L is used for length or distance instead of Δx so that it isn’t for a specific direction. Similarly, the absolute value of the PGF is calculated, and the direction is determined, by the location of High and Low pressure systems. Typically in class we won’t calculate the PGF, only the PG. However, it’ll be widely used in force balance equations.
Advection
While advection is included in the list of forces, it is actually not a true force. Still, advection can result in a change of wind speed in some locations. Wind moving through a point carries specific momentum, which is defined as momentum per unit mass. Recall that momentum is mass times velocity. Specific momentum then is simply equal to the velocity or wind speed. Therefore, as wind moves by a point, the wind can move or advect variations in winds to the fixed location.
Centrifugal Force
The centrifugal force is an apparent force that includes the effects of inertia for winds moving along a curved path. The directionality of the centrifugal force points outward from the center of the curve. The centrifugal force is the opposite of the centripetal force. As we know, inertia is the physical tendency to remain unchanged. Therefore inertia causes an air parcel to “want” to move along a straight line. Turning the air parcel along a curved path requires a centripetal force that pulls inward to the center of rotation. As a result, a net imbalance of other forces occurs.
You have felt the centrifugal force many times in your life. The centrifugal force is easily felt as you travel in a moving vehicle around a corner. The force that you feel pulling you outwards is the centrifugal force.
Coriolis Force
The Coriolis force (\(CF\)) is another apparent force that occurs due to the rotation of Earth. The Coriolis force is a deflecting force. It acts only on objects already in motion. Therefore it cannot create wind, but it can change the wind direction by deflecting it. The Coriolis force acts perpendicular to the direction of motion, but whether the Coriolis force acts 90° to the right or left of the motion vector depends on the hemisphere on Earth. In the Northern Hemisphere, the Coriolis force acts 90° to the right of the motion vector while in the Southern Hemisphere, the force acts 90° to the left of the motion vector. The equation below gives the Coriolis force
\[C F=m \cdot 2 \cdot u \cdot \Omega \cdot \sin (\phi) \nonumber \]
where \(m\) is for the mass of the object in kg, \(u\) is the speed of the object in m s-1, the symbol \(\Phi\) denotes latitude in degrees, and the angular rotation rate, \(\Omega\), is found from the rotation rate of Earth. Earth turns 2*pi radians over 24 hrs, so 2*pi/24 hrs gives \(\Omega\) = 7.27E-5 radians·s–1. Based on this, we can see that the Coriolis parameter will be 0 at the equator when \(\sin (0)=0\) and maximized at the poles when \(\sin(90)=1\).
We can also see that the Coriolis force is strongly dependent on the speed of the object. If we assume the “object” is actually wind, stronger winds will be more strongly deflected by the Coriolis force.
Turbulent Drag
Turbulent drag occurs when Earth’s surface or objects on it cause resistance to airflow and reduce the wind speed. Any object on Earth’s surface can cause drag, such as grass, trees, and buildings, which block and decelerate wind. The bottom layer of the troposphere around 0.3 to 3 km thick is called the atmospheric boundary layer (ABL). Turbulence in the ABL mixes the extremely slow movement of air near the surface with the faster movement of air in the ABL and slows the wind speed in the entire ABL.