Physics Assignment: Hydrostatic Equilibrium and Solar Insolation

This physics assignment delves into several key concepts in atmospheric physics. It begins by explaining hydrostatic equilibrium, detailing the balance between pressure gradient force and gravity, and deriving the hydrostatic equation. The assignment then utilizes a graph to estimate solar insolation values and calculate approximate temperatures using the Stefan-Boltzmann relation at different latitudes. A comparison is made with actual temperatures from Accuweather, highlighting the limitations of the Stefan-Boltzmann law due to factors like atmospheric convection and albedo. Furthermore, the assignment analyzes the temperature of air parcels based on pressure and explores Fletcher's formula to calculate hurricane speeds. Finally, it discusses the potential energy generation of a wind turbine during a hurricane.

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PHYSICS
Question 1: Explain what is meant by a fluid being in Hydrostatic Equilibrium
(a) A hydrostatic atmosphere is derived when vertical pressure gradient force acts on a vertical
direction and acceleration as a result of gravity that acts in downward direction tends to be in a
balance. The balance of forces is called hydrostatic balance otherwise hydrostatic equilibrium.
There exists a balance between the gas pressure as well as the gravitational pressure at each
layer within a stable star. This is deemed a stable equilibrium since in case the gravity was more
than the gas pressure then the star would undergo contraction. On another hand, suppose the
pressure of the gas was higher, there would be expansion of the star. In the case of stable
configuration, there has to be balance between the two. The pressure of the gas in any of the
layers is hence just equal to weight which is the gravitation al force on all the matter on top of it
just in the same way that the pressure at a given depth within a water pool is equivalent to weight
of water that is above the provided depth. This results in derivation of the term hydrostatic
equilibrium. An immediate effect is the pressure of the gas has to increase to the inside the center
of star.
The notion behind hydrostatic equilibrium is the pressure any a given point in a fluid that is at
rest is only as a result of the weight which overlies the fluid. Since pressure is a factor of force

per unit area, pressure at bottom of fluid is given by the weight of column of fluid, a unit of area
in cross section.
(b) State the names of the two key forces instrumental in maintaining hydrostatic balance in the
atmosphere. Explain briefly (in a sentence) what each of these forces do
Pressure gradient force defines the ratio of pressure difference to distance on which it acts as a
quotient of the density. It is an active force in climate system and has a direction that is
perpendicular to isobars running from low to high pressure.
Friction: Air does not tend to be very viscous hence real friction is just important in a quite thin
layer of the atmosphere next to the surface. The interaction with the surface tends to reduce the
speed of atmospheric motion.
(c) Draw a diagram to illustrate forcing acting on air parcel in hydrostatic atmosphere. With
reference to diagram, ensuring that you explain each term introduced, derive the Hydrostatic
Equation, given by dp/dx=-pg

The figure is an illustration of air parcel section as shown in the shaded region that bears a cross
section area
z defines the altitude
dz is parcel width
P and P+dp define the pressure acting up bottom and top of air parcel in that order
ρ is air parcel density
Density=Mass/Volume
Mass of air parcel having width od dz=density*cross sectional area*width
ρ∗1∗dz =ρdz

Acceleration as a result of gravity g that acts in the downward direction
ρgdz is downward force acting upon air parcel
Pressure reduces with an increase in altitude in the atmosphere hence dp which is the change in
the pressure tends to be a negative quantity
Under balance conditions:
ρgdz=−dp or
dp/dz=-ρg
Hence the equation of hydrostatic equation may be derived
Question 2: Using the graph below to estimate the solar insolation value during the month of
March assuming we can use the Stefan-Boltzmann relation to estimate temperature, what is the
approximate temperature at:
Solar insolation values for 20⁰N and 60⁰N are about 400W/m2 and 350 W/m2 in that order. For
23 March
temperature energy relation as provided by Stefan-Boltzmann law as E=KT4
In which K defines the Stefan-Boltzmann constant
Hence at 60⁰N the temperature would be:
350=5.67*10-8T4
T=7.15⁰C or 289.8K
At 20⁰N, the temperature would be:

400=5.67*10-8*T4
T=16.7⁰C or 289.8K
c) From Accuweather
Temperatures in March 23 2018 as well as 2019 for Helsinki and Honolulu were:
Helsinki-0⁰C, 3⁰C
Honolulu-27⁰C, 28⁰C
The determined temperature for Helsinki was relatively higher than actual temperature while for
the case of Hawaii is was relatively lower. Such variations may be explained by atmospheric
convection and the idea that earth is not a black substance. Stefan-Boltzmann is applicable for a
heat source incident that is purely radiant on a black substance that absorbs all emitted heat
incident. Helsinki tends to have a relatively lower content of snow and hence a higher albedo in
which it does not of reflection that absorption of the incident heat. For the case of Honolulu,
there is ladening of the equatorial winds with moisture that result in bringing in more heat as the
water is of high specific heat capacity and heat from neighbouring seas as well warm up the
islands.
Using Stefan Boltzmann equation makes limited sense since the absorption capacity of the earth
for incident heat varies in locations depending on atmosphere and geology of a place. The heat
incident for Sun is just one of numerous factors that govern temperature and not does not remain
to be the only factor hence any temperature estimated by the method may be grossly unreliable.

Question 3
Between air parcels A and B shown in the diagram below, which air parcel can you conclude is
potentially warmer? Show your work and explain your conclusion
From the temperature demonstrated in the diagram one may tell that parcel A is warmer than B.
Looking that the pressure of the two air parcels, it can be deduced that parcel A is not hotter as it
has a high pressure. Pressure decide the temperature of air parcel, the higher the pressure the
lower the temperature hence parcel B tends to be hotter than parcel A.
Question 4
Fletcher’s formula for hurricane speeds is
Vmax=16 √ Pn−Po
where Pn and Po are outer as well as inner pressures in that order. This has been provided
directly as 300 Pa
Vmax=16 √ 3=27.71 knots or 51.31 km/hr
Question 5
(a) Suppose the turbine survives during hurricane then it will generate large energy during
hurricane since the energy produced by turbine is directly proportional to speed of wind suppose
the speed of wind is higher than 12 m/s
b

(b)