In this clip I will present an overview of two fundamental concepts in meteorology hydrostatic equilibrium and the First Law of Thermodynamics.
Let's look at a small volume of air in the atmosphere represented by this cube.
Pressure is acting perpendicular to all six signs of the cube.
On the left and right sides the pressures are equal but acting in opposite directions.
Basically this means that they cancel each other out and thus there is no net force acting on the left or the right side of the cube.
The same applies for the front and the back sides no net force
however there is a pressure difference between the top and the bottom of the cube. This is because pressure decreases as we move up through the atmosphere.
Because the pressure on the top side of the cube is lower than the pressure on the bottom side of the cube there is now a net upward force acting on the volume of air.
The other vertical force acting on this volume is gravity pulling the volume down. Now if the air is static which means not moving then these two forces have to cancel each other out this is called hydrostatic equilibrium.
In an equation this reads F. G. plus F. P. equals zero. Rearranging the equation yields F. G. equals minus F.P..
For the force of gravity we can write em multiplied by G.. And for the net force due to the pressure difference we write multiplied by delta P..
Now mass is equal to volume multiplied by density where the volume can be written as area multiplied by Delta Z..
Rearranging this equation leads to row multiplied by G. equals minus delta P. over Delta Z. which is the finite difference form of the generally Quezon for hydrostatic equilibrium.
D P over D. Z. indicates how pressure is changing if we move in the positive Z. direction because rho and G. are always positive.
The right hand side of the equation is always negative. We can now conclude that in a static atmosphere pressure always decreases with height. And moreover the pressure decrease is proportional to the density the higher the density the more rapid the pressure decrease
near the earth's surface the pressure decrease is approximately one.
Per eight meters as density becomes lower at higher elevations the pressure decrease also diminishes as we move up through the atmosphere
the First Law of Thermodynamics is actually a statement about the conservation of energy. To begin with we need to know what internal energy refers to.
A gas consists of particles atoms and or molecules moving at high speeds and colliding with each other and the walls
therefore each particle has a certain amount of kinetic energy.
This some of the kinetic energy of all of these particles is called the internal energy which we indicate with a U..
The internal energy is proportional to the temperature the higher the temperature. The faster the particles move thus the higher their kinetic energy and internal energy become.
Let's assume that we have a unit of mass of a gas contained in a volume such as the gas in the cylinder below the piston. We keep the volume constant by keeping the piston fixed. Now if we supply a certain amount of energy D.E.Q. to this gas then all of this energy is used to increase the internal energy of the gas. In an equation the energy per unit of mass D. Q. equals D. you if the internal energy increases then the temperature of the gas will also increase. So D. you is proportional to D. T. multiplied by a constant c v which is the specific heat of the gas at a constant volume.
For dry air with no water vapor the specific heat at a constant volume equal seven hundred and seven thousand Joules per kilogram per Kelvin
if the piston is allowed to move without friction then the pressure of the gas is always exactly the same as the pressure outside the cylinder so now we have a situation where the pressure is kept constant in this situation there is not only a change in the internal energy but the gas will also perform work.
First let's look at how we can calculate the amount of work
the amount of work is defined as the force multiplied by the distance over which the forces acting.
If the piston with Area A is pushed outward along a distance as then the amount of work performed by the gas D. W. becomes P. multiplied by a multiplied by S.
where a multiplied by S. is just the increase in volume D V
in a P.T. V. diagram the amount of work is equivalent to the colored area under the eyes of Barack line A B..
If we want to refer to the amount of work per unit mass we will use a lowercase W. which can now be calculated as D. W. equals P. multiplied by D. Alpha. Where Alpha is the volume per unit of mass or specific volume Alpha is thus defined using the following formula Alpha equals V. divided by M. which is equal to one divided by rho.
The total amount of energy per unit mass D. Q now becomes the sum of the change in internal energy which is the first term here on the right hand side of the equation and the amount of work which is the second term on the right hand side
combining the first law of thermodynamics with the gas law will result in an alternative expression D.Q. equals C P multiplied by D. T. minus Alpha multiplied by D.P. where C.P. is the specific heat of dry air at a constant pressure it can be shown that C P equals c v plus R. v with this equation it is easy to determine that C P equals one thousand and four Joules per kilogram per Calvin.
Next we will consider what is called an addy a bad a process this is a process whereby the amount of energy put into or taken out of the system equals zero. In other words there is no exchange of energy between the system and its environment.
In the atmosphere we imagine a certain air parcel as a system and the environment is the rest of the atmosphere.
It is difficult to imagine that no exchange takes place in the atmosphere since there are so many processes exchanging energy. These processes are called diabetic processes two examples of diabetic processes are turbulent mixing where air from the environment is mixed into the parcel or air from the parcel is mixed into the environment and active exchange of energy takes place.
Another diabetic process could be the absorption or even mission of radiation by the air parcel these two processes will supply or remove energy from an air parcel.
To exclude diabetic processes we will consider air parcels that have a large spatial scale when this is the case the only mixing exchange occurs far away
we will also only look at relatively short timescales.
Then the rather slow absorption or emission processes do not have enough time to act
assuming that we are dealing with an atty about a process the first law of thermodynamics reduces to this expression where D.Q. is now equal to zero by definition
combining this equation with hydrostatic equilibrium reveals the following relationship D.T. over D.Z. is equal to minus G. divided by C. P. which is equal to minus gamma D. where gamma D. is called the dry adiabatic lapse rate it shows that during an atty about a process if an error parcel moves up or down that the temperature change with height is a constant and equals approximately ten Calvin per kilometer.
This concept allows us to introduce the potential temperature feta it represents the temperature that an air partial would have if it was brought Addie a bad eclipse from a certain pressure level P one to the reference level of one thousand hectare back al
the formula for theta can be found if we combine the addy about a form of the First Law of Thermodynamics with the gas law and integrate from a certain pressure level P. one where the temperature equals T one two reference pressure of one thousand hectare Pascoe
potential temperature is a useful concept in order to have a fair comparison between air parcels that are situated at different levels in the atmosphere.
Consider two air parcels one at a height of three kilometers where the pressure equals seven hundred hectare Pascoe and the temperature is zero degrees Celsius. Another air parcel is located at a height of one kilometer where the pressure equals nine hundred hectare Pascal and the temperature is ten degrees Celsius.
This latter air parcel has the highest temperature but is located at a lower altitude if we bring both parcels to the reference level of one thousand hectare Pascale then both of them will have an increase of ten degrees per kilometer indicated by the red lines which are called dry adieux bats.
At one thousand hectare Pascal It appears that the upper parcel has a higher temperature a higher potential temperature than the lower partial
using potential temperature enables us to make a fair comparison between air parcels at different pressure levels that is without the effect of the differences in height.
If the atmosphere is well mixed than the potential temperature is almost the same at all vertical levels from these observations made at several different heights along a two hundred meter high tower we can see that during the day the temperatures at all levels may be different but the potential temperatures are almost equal when mixing stops for example during the night all temperatures will diverging again.