2. Briefly describe in your own words the physical meaning of each of the terms in thefollowing form of the thermodynamic energy equation:ӘТ ӘТ ӘТ+u- +v. - Spw=Ət əx Əy=1 dQCo dtT 20where Sp- What is the vertical coordinate for this equation? Define w andдрgive a mathematical expression for it. Explain the physical signficance of S, and theterm -Spw? What form of this equation would apply for adiabatic flow?

Answered: Briefly describe in your own words the…

Applications and Investigations in Earth Science (9th Edition)

9th Edition

ISBN:9780134746241

Author:Edward J. Tarbuck, Frederick K. Lutgens, Dennis G. Tasa

Publisher:Edward J. Tarbuck, Frederick K. Lutgens, Dennis G. Tasa

Chapter1: The Study Of Minerals

Section: Chapter Questions

Problem 1LR

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Question

2. Briefly describe in your own words the physical meaning of each of the terms in the
following form of the thermodynamic energy equation:
ӘТ ӘТ ӘТ
+u- +v. - Spw=
Ət əx Əy
=
1 dQ
Co dt
T 20
where Sp
- What is the vertical coordinate for this equation? Define w and
др
give a mathematical expression for it. Explain the physical signficance of S, and the
term -Spw? What form of this equation would apply for adiabatic flow?

Expert Solution

Step 1

In the equation, each term represents, the time derivative of temperature in terms of constant pressure.

The first term is simply representing the partial change in temperature with time.

The second and third term represents the rate of change of temperature in terms of horizontal velocities u and v.

u and v can be written as,

u = \frac{d x}{d t}

And,

v = \frac{d y}{d t}

The fourth term defines the temperature change with time in terms of the stability parameter S_p and the vertical velocity term ?. It is negative because, it maintains the balance between the horizontal and vertical components.

The 5^th term in the equation (right hand side) represents the time rate of change of temperature in terms of heat change with time and specific heat at constant pressure cp.

The physical significance of the terms in the equation are given here

For this equation, the vertical coordinate is nothing but, pressure (p).

Pressure has been used replacing height as vertical coordinate.

The term ? is defined as the vertical velocity in case of pressure as vertical coordinate.

It can be written as,

𝜔 = \frac{d p}{d t}

The term S_p is the static stability parameter or stability measure of the equation.

It is defined as,

θθ

S_{p} = - \frac{T}{θ} \frac{\partial θ}{\partial p}

Here, ? = Potential temperature

p = pressure.

In other words, it is the measure of the temperature changes with respect to the pressure coordinate. But using pressure as vertical coordinate, the value of the vertical coordinate at the lower boundary is no longer fixed in time and the static stability factor is strongly varying in the vertical.

The term, ?*_Sp is simply measures, the deflection in temperature with respect to (over the) the vertical coordinate.