In this clip I will illustrate the theory of juice trophic motion by using the example of curved isobars
We will also examine the theory of circular motion and learn how to derive the gradient wind relationships
First let's review the theory of uniform circular motion.
If a mass is in a circular orbit.
Then in a given time delta T. it will cover a part of the circle delta S..
If the radius of the circle is denoted by R. T.. And the angle is denoted by Delta alpha
then delta S. is equal to R. T. multiplied by Delta Alpha. So the velocity v of the mass is R. T. multiplied by Delta Alpha divided by Delta T. or R T multiplied by Omega where omega is the so-called angular velocity expressed in radians per second
according to Newton's Laws a mass will only change its velocity if there is a force acting upon the mass in this case only the direction of the velocity changes. The corresponding force is directed to the center of the circle and is called the centripetal force F. C.
it can be shown that F. C. is equal to mass multiplied by velocity squared divided by R. T.
which is equal to mass multiplied by angular velocity squared multiplied by R. T..
At this point I need to introduce the natural coordinate system.
Instead of using a predefined X. and Y. direction. We will use coordinates that are determined by the local direction of the flow that is by the local velocity v
the first coordinate is pointing in the same direction as the local velocity it is called the S. direction where S. stands for stream wise or along the local velocity
the second cordon and is perpendicular to the local velocity or as we say normal to the local velocity it is called the N. direction where N. stands for normal.
As we have two possibilities here we have chosen to have the positive end direction pointing to the left of the local velocity
Now if the trajectory of the mass is curved We also need the local radius of curvature R. T.. Which is positive for cyclonic motion and negative for anti cyclonic motion
let's start with the circular motion surrounding a psych loan. That is a low pressure system in this case V. is turning counter-clockwise. And the end direction is pointing to the center of the cycler.
For the pressure gradient in the natural coordinate system we now have D. P. over D. N.. Its value is negative since pressure decreases in the positive Endor action
this is a cyclonic case and thus our T. is positive
now there are two horizontal forces acting on this air parcel. The pressure gradient force is by definition pointing to decreasing pressure.
The Coriolis force is pointing to the right of the velocity.
The sum of the pressure gradient force and the Coriolis force must be a force pointing to the center of the circle therefore the Coriolis force must be smaller than the pressure gradient force
the vector sum of these two forces is called the centripetal force.
Note that this is not an extra force but just the sum of the other two forces
algebraically we can say that the centripetal force F. C. equals the pressure gradient force minus the Coriolis force this leads to this quadratic equation for V.
If we solve this equal the expression for a V. emerges.
This philosophy is called the gradient wind speed
the square root consists of two terms.
And it is easy to see that both terms are positive.
It means that there will always be a solution to this equation no matter how large the pressure gradient D.P. over D. N. becomes.
For instance in tropical cyclones and tornadoes the value for the pressure gradient may become very high
in the case of a high pressure system we have anti cyclonic circular motion v turning clockwise. And the end direction is pointing away from the center of the cycler.
Yet again D.P. over D. N. is negative as pressure decreases in the positive and erection
because this is an anti psych lone R T as negative
once again the pressure gradient force points towards the decreasing pressure that is away from the center of the anticyclone.
The Coriolis force is directed to the right of the velocity this time towards the center of the circle.
The vector sum of these two forces acts as the centripetal force.
Algebraically we can say that the centripetal force F C equals the Coriolis force minus the pressure gradient force.
This leads to a similar quadratic equation for V.
Solving this equation leads to the following expression for the gradient wind around an anti Cyclon.
Please note that the only difference between the expression for the gradient wind around the cycle and the expression for the gradient wind around and anticyclone is the minus sign before the square root.
Again the square root consists of two terms here the first term is positive but the second term is negative it means that a solution to this equation will not exist if the pressure gradient D.P. over D. N. becomes too big it implies that the pressure gradient near an anti psyco on cannot exceed a certain limit as a result high wind speeds do not occur near anticyclones
in order to calculate the gradient wind at a certain location P. from the data on a regular surface map with isobars you need to follow certain steps. First fit a part of a circle along the ice O'Barr through point P..
Then draw two perpendicular lines at the ends of the circle segment
where they intersect is the center of the circle.
Note that this is usually not the same point as the center of the system the radius of curvature R. T. is equal to the radius of the circle.
Determine the perpendicular distance Delta N. between two nearby isobars.
Then determine the pressure difference delta between these two nearby isobars.
Finally calculate the gradient wind after calculating the density from temperature and pressure at point P. with the gas law
when dealing with identical ice of our distances the following three situations can be considered anticyclonic straight and cyclonic curvature.
For all three of these situations the pressure gradient force is the same. However the Coriolis force and hence the wind speed must be different in all three cases.
From this comparison it is obvious that an anti cyclonic curvature and hands as the wind speed and a cyclonic curvature decreases the wind speed relative to the geostrategic wind.