The equation of state for an ideal gas is:

$$P = \rho R_d T$$

Where:

- $$P$$ is the pressure

- $$\rho$$ is the density of the air

- $$R_d$$ is the specific gas constant for dry air (287.058 J/(kg K))

- $$T$$ is the absolute temperature

2. Hydrostatic Equation:

The hydrostatic equation describes the vertical variation of pressure in the atmosphere:

$$\frac{dP}{dz} = -\rho g$$

Where:

- $$dP/dz$$ is the vertical pressure gradient

- $$g$$ is the acceleration due to gravity (9.80665 m/s^2)

3. Equation of Motion:

The equation of motion for the atmosphere is given by the Navier-Stokes equations, which describe the balance between forces acting on an air parcel. In simplified form, the horizontal equation of motion is:

$$u\frac{\partial u}{\partial x} + v\frac{\partial u}{\partial y} + w\frac{\partial u}{\partial z} = - \frac{1}{\rho}\frac{\partial P}{\partial x}$$

Where:

- $$u, v, w$$ are the wind components in the x, y, and z directions, respectively

- $$P$$ is the pressure

4. Continuity Equation:

The continuity equation expresses the conservation of mass and states that the divergence of the velocity field is equal to zero:

$$\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} + \frac{\partial w}{\partial z} = 0$$

These four equations form the basic set of equations used in atmospheric modeling and weather prediction. They describe the physical laws governing the behavior of the atmosphere and are solved numerically to simulate and understand atmospheric processes.