“In 2015 winter, it was very cold. I wondered where the source of the cold was. How powerful it was! Such that it swept the whole of North America, Europe, and Asia! I tried to find the “wind equation”, or “wind speed equation” on the net. However, nothing could be found on the net. It was well known that “moving air forms wind” was common sense in primary school. Why not thing can be found? I told myself that perhaps this is a very difficult problem, because the air has no shape, no boundary, and no volume. How to set up an equation? That was easy, I replied to myself. Giving the air wrapped by a “zero-weighted membrane”, then, it has shape and volume and thus scientific laws can be used. Meanwhile, it didn’t change the stress field, since there was nothing wrapped in the air. Based on this idea, I published two papers [1] [2] . That is my story on “Wind Speed Equation” (see “preface” of [3] ).

The “Wind Speed Equation” (simple form) (or called “wind-temperature equation”):

$\stackrel{\dot{}}{u}=k\nabla T$, $\left(k=c/m=\text{const}\right)$ (2-1)

where $u$ = wind speed, $\nabla =i\frac{\partial }{\partial x}+j\frac{\partial }{\partial y}+k\frac{\partial }{\partial z}$, Hamilton operator, $T$ is temperature, k, c, m (mass) are constants.

The solution of (2-1) is:

$u\left(s,t\right)=\text{exp}\left[k\left(s-t^2\right)\right]$, (2-2)

where $u$ = wind speed, $s=xi+yJ+zK$ = position vector.

(2-1) is a PDE (Partial Differential Equation). It states that the derivative of wind speed respective to time is proportional to the derivative of temperature with respect to trace. This result can be viewed as a rule that strong wind companies with temperature sharp dropping. Or, temperature sharp differed between positions in the track, causing strong wind. Because it has a simple form, suits for wide use. Its method has copied to vertical type, to rotation air (circulation cyclone), etc. Thus, it becomes a base for the description of certain types of wind.

Wind is invisible. How can you prove the correctness of (2-1)?

Confirm the truth by fact, comparing,

The motion equation of [4] , derived by different models based on modifying the Naiver-Stoles equation, with the result well agreed with (2-1).

The vertical type of wind speed equation is shown as:

$\stackrel{\dot{}}{u}=\nabla \left(p/{\rho }_{air}\right)=k_z\nabla p$, $\left(k_z={\rho }_{air}^{-1}\right)$ (3-1)

where $u$ = wind speed, $k_z$ = constant, depending on z (highness).

(3-1) means that the derivative of wind speed $u$ with respect to time t proportions to the derivative of $\left(p/{\rho }_{air}\right)$ respect to track (space).

The solution of (3-1) is:

$u\left(s,t\right)=\text{exp}\left[k_z\left(s-t^2\right)\right]$, $\left(k_z={\rho }_{air}^{-1}\right)$, (3-2)

where $u$ = wind speed, $s=xi+yJ+zK$ = position vector.

Traditionally. methods, tools for calculation of ${\rho }_{air}={\rho }_{air}\left(p,z\right)$ as functions of p and z [6] [7] , based Boyle’s law and Charles law for static description. i.e., they have no connected wind speed. However, our treatment of ${\rho }_{air}$ is different from that of traditional. It connects with wind speed by “ $w-p/{\rho }_{air}$ equation” (3-1), based on Boyle’s law, Charles law and together with the Newton’s second law. The calculation of ${\rho }_{air}$, can be found in Appendix of [5] .

The traditional method obtains static (fixed) results, while our method gets variable results. Both method have their use for different goals.

Rotating air is a natural phenomenon seldom seen. Sometimes, it can be seen in buildings' corners, but it soon disappears. Only strong rotating wind can last.

Mathematical description of rotating motion [9] .

$rots=\nabla \otimes s$,(4-1)

where $s$ = position vector. $\otimes$ = vector product, or outer product.

Use (4-1) to (2-1), we have Rotation Equation.

$Rot\left(m\stackrel{\dot{}}{u}\right)=Rot\left(cT\right)$, (4-2)

The necessary and sufficient conditions for existence of (4-2) is:

$s\ne 0,s\ne I,s\ne \nabla ,I=\text{unit}\text{vector}$, (4-3)

If

$s\parallel \nabla ,\text{or}s\parallel I,\text{or}s=0,$ (4-4)

then, $Rots=\nabla \otimes s=0$. i.e., Rotation disappears.

Using the method similar to the wind speed equation but for cylindrical coordinates and for UCM (uniform circular motion), of rotation around z-axis, calculated by method of section. We got two set non-linear PDE of a point in UCM of steady flow. These PDFs to be solved.

The above paper obtained PDF with 4 unknown functions, 2 independent variables, used the method of dimension analysis and method of separating variables. Complimentary equation derived by iteration method. After two round iterations, using g-contraction mapping theorem for stopping iteration. We got 4 solutions (2-51 - 2.54).

Application of (2-51 - 2.54). We have something useful. Maximum wind speed is located on the edge of the cyclone $r_1$, minimum wind speed is located on the center of the cyclone. Staying at the center of the cyclone should be safe. Other solutions have a similar distribution.

The trip starts from $A\left(x_0,y_0,z_0,t_0\right)=A\left(0,0,0,0\right)$,

$\begin{array}{l}A\left(x_0,y_0,z_0,t_0\right)\stackrel{}{\to }{d}_1\stackrel{}{\to }B\left(x_0,y_0,z_1,t_1\right)\stackrel{}{\to }{d}_2\stackrel{}{\to }C\left(x_1,y_1,z_1,t_2\right)\\ \stackrel{}{\to }{d}_2\stackrel{}{\to }D\left(x_1,y_1,z_0,t_3\right)\stackrel{}{\to }{d}_1\stackrel{}{\to }E\left(x_0,y_0,z_0,t_4\right)\end{array}$(5-1)

where $d_1=\left|z_1-z_0\right|=z_1=\text{distance}$, $d_2=\sqrt{{\left(x_1-x_0\right)}^2+{\left(y_1-y_0\right)}^2}=\sqrt{x_1^2+y_1^2}$. Since the expression of position (highness) $z_1$ is different. In our formula (3-2) is variable, while in traditional expression the highness is fixed. It suits for use as a standard. Therefore, we use $z_1$ to represent the standard form of highness by red color.

In (5-1), we show a round trip of one year. It starts from A. $t_0$ = 0 = 2023/06/01. The trip ended in E. $t_4$ = 2024/06/01. We set $z_1$ = 100 km. Dividing 100 Km into 10 equal units, i.e., $z_1$ (9/10) - 90 km. etc. Using (3-2) for 10 units, one can calculation $t_1$ by the sum of 10 unit’s. Similarly, calculation for $t_2$, $t_3$, $t_4$. Note that $t_1\ne t_3$, and $t_2$, $t_4$ use (2-2).

During this one-year trip, the sun played an important role. In summer, it heats the sea level hot, and hot air rises up. But in winter, without the sun heating. The sea level becomes colder, and air drops down from the sky to sea level.