Analysis of Hydrostatic Balance in Climate Models, Spring 2020

This assignment explores the concept of hydrostatic balance in atmospheric conditions, detailing the equation that expresses this balance and explaining its limitations in climate models. The solution elaborates on the hydrostatic equilibrium, the forces involved, and the mathematical representations. Furthermore, the assignment delves into energy balance models, specifically examining how a single homogenous atmospheric layer can illustrate the greenhouse effect. The solution analyzes the provided equations, focusing on energy emitted from the Earth's surface and the impact of atmospheric transparency on solar and infrared radiation. The assignment covers key elements of climate modeling, including hydrostatic balance, energy balance, and the factors affecting these processes.

Solution.1
(1) The environment is mainly in equilibrium and hydrostatic balance, when it is between
the upward directed force and the downward directed force of gravity ("The
Hydrostatic Equation").
This situation can be expressed by the following equation:
∂p/∂z = –ρg (1)
Where, ∂p/∂z = partial derivative of p with respect to pressure, z and p.
(2) The principle of hydrostatic equilibrium can be elaborated as the pressure at any point
in the fluid at rest condition.
The pressure can be defined as the force /unit area. The pressure at the bottom of the
fluid is the weight of the column of the fluid, which is one unit of the area in cross-
section.
If the fluid is in the form of incompressible, then the density will be independent of
the pressure and the weight of a liquid column is just proportional to the height of the
liquid.
The pressure can be elaborated by this formula shown below:
P = g ρ h
Where,
P = Pressure

h = height
ρ = density
For an environment in the condition of hydrostatic equilibrium, the balance of forces
in the vertical is shown below,
−δp = gρ δz
In the limit of,
δz → 0, ∂p/∂z = −gρ
It is hydrostatic equation. Here, the (-) sign represents that the pressure minimizes
with the increasing height ("The Hydrostatic Equation").
(3) There are so many limitations of hydrostatic balance assumption in the climate
change, some of them are shown below:
 Horizontal length is higher than the vertical length
 Bed friction and viscous shear stresses on the fluid components
 Implementation of roughness on the grid plane not on the grid walls
 Macro effect variations at various parameters such as, direction, and channel
shape etc.
Solution.3
(a) ∂u
∂ t =−u ∂u
∂ x −v ∂u
∂ y −w ∂u
∂ p +fv−g ∂ z
∂ x +Fx
The red terms in the right hand side of the equation shows the u-momentum equation
in terms of magnitude in corresponding with x axis and y- axis.
The green term shows the u-momentum equation in corresponding with pressure
change as δp.
In the blue term equation at R.H.S, term fv can be defined by the equation shown
below:
Cx = fv
Cy = -fu
These are the component balance, where Cx depends on v and Cy depends on u.
Here, f is the Coriolis factor (Kushnir, "Laws of Atmospheric Motion and Weather")
Fx = - ΔpΔyΔz = (ρΔpΔyΔz)a, where ρ is the density of air.
-g. ∂z/∂x = hydrostatic equation in terms of gravity

(b) Scale analysis for the vertical momentum equation proceeds with the equation, where P
is shown as the pressure’s vertical variation = 1000mb or 105Pa.
This analysis represents that the gravity and the pressure gradient terms are dominant.
In case of, Synoptic scale analysis of the horizontal momentum, various terms are very small
which can be ignored without any loss in accuracy. Hence, various terms can be ignored
such as, coriolis terms, viscous terms and the curvature terms. Hence, this analysis shows the
pressure gravity and gravity terms as less dominant ("SCALE ANALYSIS OF THE
MOMENTUM EQUATIONS")
References

SCALE ANALYSIS OF THE MOMENTUM EQUATIONS. (n.d.). Retrieved March 30,
2020, from
http://snowball.millersville.edu/~adecaria/ESCI342/esci342_lesson06_scale_analysis.
pdf
The Hydrostatic Equation. (n.d.). Retrieved March 30, 2020, from
https://maths.ucd.ie/met/msc/fezzik/Phys-Met/Ch03-Slides-2.pdf
Kushnir, Y. (n.d.). Laws of Atmospheric Motion and Weather. Retrieved March 30, 2020,
from https://eesc.columbia.edu/courses/v1003/lectures/general_circulation/images/
Kushnir_Lecture4.pdf