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--- un:cloud [2025/08/03 12:43] – created asad
+++ un:cloud [2025/08/29 10:02] (current) – asad
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-====== Cloud ======
+====== Clouds ======
+For the health of any planet, clouds are very important. The clouds of Earth regulate its [[albedo]], because clouds reflect more sunlight than ocean or land. The percentage of Earth's surface covered by clouds determines what percentage of sunlight is reflected, meaning how cool Earth remains.
+{{https://upload.wikimedia.org/wikipedia/commons/thumb/5/57/Cloud_types_en.svg/1024px-Cloud_types_en.svg.png?nolink}}
+Above are shown the 10 main types of clouds in Earth's [[atmosphere]]. At low altitude are found cumulus, stratus, and stratocumulus. At mid altitude are altocumulus and altostratus. At high altitude are cirrocumulus, cirrostratus, and cirrus. In addition, cumulonimbus clouds can be found at all heights, and nimbostratus can exist from low to mid altitudes. Cumulus can also be found in the upper layer. And now, through physics, we will see how these fluffy cumulus clouds are formed.
+Suppose a parcel of gas of unit mass ($m=1$ kg) rises upward in an adiabatic process, meaning no heat flows in or out of it. This gas must obey the first law of thermodynamics which states,
+$$ dQ = C_VdT+pdV $$
+where $Q$ is heat, $P$ is pressure, $V$ is volume, $T$ is temperature, and $C_V$ is its specific heat capacity at constant volume. For an adiabatic process, $dQ=0$. Now, from the ideal gas law we can write
+$$ PV = nRT = \frac{m}{M_m}RT = \frac{RT}{M_m} $$
+where $n=m/M_m$ is the number of moles, $m$ is the total mass of the gas, and $M_m$ is the molar mass or the mass per mole. Even if pressure, volume, or temperature change slightly ($\delta$), this equation remains true, meaning
+$$ (P + \delta P)(V + \delta V) = \frac{R(T + \delta T)}{M_m} $$
+where if we ignore small terms ($\delta P\delta V$) and subtract the previous equation from this, we get the following form.
+$$ P\,dV + V\,dP = \frac{R}{M_m} dT $$
+Here the symbol $\delta$ has been replaced with $d$ to denote infinitesimal changes. Combining this equation with the first law of thermodynamics, we can write,
+$$ 0 = C_V\,dT + \left( \frac{R}{M_m} dT - V\,dP \right) = C_P\,dT - V\,dP $$
+because $C_P-C_V=R/M_m$, where $C_P$ is the specific heat capacity of the gas at constant pressure. When this gas rises, its pressure must equal the surrounding atmospheric pressure, otherwise the gas would rapidly expand or contract to establish pressure balance. This means the gas follows the equation of [[hydrostatic-equilibrium]], i.e., $dP=-\rho g\,dz$. Combining this with the earlier equation, we get
+$$ C_P\,dT + V\rho g\,dz = 0 \Rightarrow \frac{dT}{dz} = -\frac{g}{C_P} $$
+because $V\rho=1$, since the gas is of unit mass. This is the temperature gradient of the gas, known as the adiabatic lapse rate. The way temperature decreases from the surface upward through the troposphere of many planets can be explained with this equation. In the case of Earth, however, the latent heat released when vapor condenses into clouds must be taken into account, though that is a small correction. Roughly, within 10 km this gradient holds true. Its explanation is as follows.
+Any derivative $dy/dx$ can be visualized as a plot, with $x$ on the horizontal axis and $y$ on the vertical axis. For the adiabatic gradient, if we plot height on the x-axis and temperature on the y-axis, we get a straight line with a negative slope, meaning temperature decreases with height. The slope (gradient) is determined by gravity and heat capacity.