There are three real forces important for atmospheric motion:

- Gravitational Force
- Pressure Gradient Force (PGF)
- Friction (really drag on the flow by turbulence)

Hence we can sum these real forces:

$\sum \overrightarrow{{F}_{a}}=\overrightarrow{{F}_{g}}+\overrightarrow{{F}_{p}}+\overrightarrow{{F}_{f}}$

We put the subscript "*a*" on these forces to indicate "absolute" because they are true in an inertial reference frame. Thus, in the absolute reference frame,

$$\frac{{D}_{a}{\overrightarrow{U}}_{a}}{Dt}=\frac{\sum \overrightarrow{F}}{m}$$

Let's examine each of these real forces in more detail.

### 1. Gravitational Force

Recall that the gravitational force on a mass *m* is simply the weight of the mass, which is given by:

$$\begin{array}{l}{\overrightarrow{F}}_{g}=m\overrightarrow{g}*\\ where\\ \overrightarrow{g}*=-\frac{GM}{{r}^{2}}\overrightarrow{k}\end{array}$$

where *M* = Earth’s mass (5.972x10^{24} kg), *r* = distance from Earth’s center (6.371x10^{6} m* +* z, where *z* is the height above Earth's surface), and *G* = gravitational constant (6.67384 × 10^{-11} m^{3} kg^{-1} s^{-2}). Ignoring the minor effects of topography and the horizontal variation of Earth's density, the real gravitational force points directly towards Earth’s center, right down the *k* vector.

The specific real gravitational force is $\overrightarrow{g}*$.

### 2. Pressure Gradient Force (PGF)*

*This is actually the specific pressure gradient force, giving dimensions of L/t^{2}.

The derivation is similar to what we have already done in Lesson 2.2 to find hydrostatic equilibrium, except that we will look at only the pressure forces in this case. It will serve as a quick review. Consider the x-direction first:

$m\frac{du}{dt}=p\left(x\right)A+\left(-p\left(x+\Delta x\right)A\right)=p\left(x\right)A+(-\left(p\left(x)+\Delta p\right)\right)A$

$m\frac{du}{dt}=\frac{A\Delta x(p\left(x\right)-p\left(x)-\Delta p\right)}{\Delta x}=-V\frac{?p}{\Delta x}\approx -V\frac{\partial p}{\partial x}$

$\frac{du}{dt}=-\frac{V}{m}\frac{\partial p}{\partial x}=-\frac{1}{\rho}\frac{\partial p}{\partial x}$

We can also show the acceleration due to the pressure gradient force in the other two directions to get the equation:

$\frac{d\overrightarrow{U}}{dt}=-\frac{1}{\rho}\overrightarrow{\nabla}p=-\frac{1}{\rho}\left[{\overrightarrow{\nabla}}_{H}p+\overrightarrow{k}\frac{\partial p}{\partial z}\right]$

$$\text{where}\overrightarrow{U}=\overrightarrow{i}u+\overrightarrow{j}v+\overrightarrow{k}w\text{and}{\overrightarrow{\nabla}}_{H}p=\overrightarrow{i}\frac{\partial p}{\partial x}+\overrightarrow{j}\frac{\partial p}{\partial y}$$

We separate the horizontal motion from the vertical motion because we will see that gravity is a dominant force in the z-direction that does not have an equivalent in the x or y directions. For instance, a typical horizontal pressure gradient is 10 hPa over 1000 km, with an air density of 1.2 kg m^{-3}, which gives 0.0008 m s^{-2} for the acceleration of the horizontal wind. This value is much less than the acceleration due to gravity, which is 9.8 m s^{-2}. Thus, we can write the equations of motion as:

$$\frac{d\overrightarrow{V}}{dt}=-\frac{1}{\rho}{\overrightarrow{\nabla}}_{H}p\text{and}\text{\hspace{0.17em}}\frac{\partial w}{\partial t}=-\frac{1}{\rho}\frac{\partial p}{\partial z}$$

where$\overrightarrow{V}=\overrightarrow{i}u+\overrightarrow{j}v$is the horizontal velocity.

#### Example

Let's do a quick calculation of the pressure gradient force from a map of surface pressure on 26 June 2015:Note that Pennsylvania is 250 km from its southern border to its northern border.

The following video (1:20) will explain the process:

### 3. Friction

We can think of friction as being processes that impede the air flow. There are a few different ways that this happens in the atmosphere.

On small scales, less than a centimeter, air flowing past slower moving air encounters resistance at the molecular scale that is called **viscosity**, which is designated as µ for **dynamic viscosity**. So, for example, if you think of air near a surface, the air at the surface is stationary. By viscosity, it slows that air above it, just as that air slows the air above it. In general, viscosity acts to slow air that is moving past slower moving air, ultimately converting the mechanical energy of the air flow into the random kinetic energy of heating. We show without derivation that the viscous force, F_{r}, is given by:

$${\overrightarrow{F}}_{r}\cong \frac{\mu}{\rho}\text{\hspace{0.17em}}{\nabla}^{2}\overrightarrow{U}=v\text{\hspace{0.17em}}{\nabla}^{2}\overrightarrow{U}$$

where* v* is the kinematic viscosity and ${\nabla}^{2}=\overrightarrow{\nabla}\xb7\overrightarrow{\nabla}=\frac{{\partial}^{2}}{\partial {x}^{2}}+\frac{{\partial}^{2}}{\partial {y}^{2}}+\frac{{\partial}^{2}}{\partial {z}^{2}}$ is called the **Laplace operator** or the **Laplacia**n, and $\overrightarrow{U}$ is the air parcel velocity. The viscous force is important for resisting flow and dissipating air flow on small scales, but it is not an important force on larger scales when compared to other forces such as gravity and the pressure gradient force. Therefore, we will delay any more discussion of viscosity until Lesson 11, which is devoted to the atmospheric boundary layer.

There is a resistance to flow that is important for larger scale atmospheric motion, even synoptic scale, and we will discuss its origins here. The flow in the atmosphere's lowest kilometer or two, called the atmospheric boundary layer, is often turbulent, with chaotic large and small swirls of air that, when taken together, have momentum in all directions. During the day, turbulence is generated by convection. As convection bubbles up during the day, this turbulence extends to the top of the planetary boundary layer, where the air flowing horizontally through the upper boundary layer encounters it. During the night, turbulence is also generated by windshear throughout the boundary layer. No matter how turbulence is generated, it provides a drag on the horizontal flow at the top of the boundary layer. This turbulent drag generally reduces the horizontal wind speed, which is why it is often referred to as friction, even though the word "friction" really applies only to molecular scale interactions such as viscosity or resistance to motion of one body in contact with another.

For a boundary layer with convection, the turbulent drag can be written approximately as the negative of the product of four terms: the dimensionless drag coefficient, *C _{d}*, divided by the planetary boundary layer height,

*h*, multiplied by the magnitude of the horizontal velocity, $\left|\overrightarrow{V}\right|=\sqrt{{u}^{2}+{v}^{2}}$, and also the horizontal velocity:

${\overrightarrow{F}}_{f}=-\frac{{C}_{d}}{h}\left|\overrightarrow{V}\right|\overrightarrow{V}$

Even though this turbulent drag is not really friction, it is an important resistance to the flow on large scales in the upper boundary layer and so we will keep it, and not viscosity, as the friction term in the momentum equations.

#### Inertial (Real) Force Summary

The real forces can be summarized in the following equations. The first equation is in vector form; the three equations following are the same as in the vector equation, but the three equations represent the equations of motion in the x, y, and z directions.

$\frac{D{\overrightarrow{U}}_{a}}{Dt}=-\frac{1}{\rho}\overrightarrow{\nabla}p+\overrightarrow{g}*-\frac{{C}_{d}}{h}\left|\overrightarrow{V}\right|\overrightarrow{V}$

$\frac{\partial u}{\partial t}=-\frac{1}{\rho}\frac{\partial \text{p}}{\partial x}-\frac{{C}_{d}}{h}\left|\overrightarrow{V}\right|u;\frac{\partial v}{\partial t}=-\frac{1}{\rho}\frac{\partial \text{p}}{\partial y}-\frac{{C}_{d}}{h}\left|\overrightarrow{V}\right|v;\frac{\partial w}{\partial t}=-\frac{1}{\rho}\frac{\partial \text{p}}{\partial z}-g;$