oceanic and atmospheric fluid dynamics · with me on this project. my gratitude also goes to...

Bachelor Thesis

zur Erlangung des akademischen Grades Bachlorin in der

Studienrichtung Technische Mathematik

Oceanic and Atmospheric FluidDynamics

Author:

Michaela Lehner

Supervisor:

Dr. Peter Gangl

28. Oktober 2017

Abstract

A system of partial dierential equations called the primitive equations of the oceanand the atmosphere, describing the behaviour and the properties of those uids, isfundamental when studying and predicting the behaviour of atmospheric and oceanicuids. The most common eld of application for the primitive equations is the mod-elling of the ocean and the atmosphere, as well as the description of their behaviourand predictions building on that. Many other models mathematically describing theocean and the atmosphere are derived from the primitive equations, making them thebasis of discussions concerning the ocean and the atmosphere. Goal of this thesis wasthe derivation of the primitive equations for the ocean and the atmosphere from thegeneral physical equations and to consider boundary and initial conditions and somespecial settings.

i

Zusammenfassung

Fundamental beim Studieren und Vorhersagen des Verhaltens von atmosphärischenund ozeanischen Fluiden ist ein System partieller Dierentialgleichungen, das das Ver-halten und die Eigenschaften des Ozeans und der Atmosphäre genau beschreibt. Diesewerden als die Primitiven Gleichungen des Ozeans und der Atmosphäre bezeichnet .Zugrunde liegen ihnen die physikalischen Eigenschaften, die diese kompressiblen Fluideausmachen. Das wichtigste Anwendungsgebiet für diese Gleichungen ist die Modellie-rung des Ozeans und der Atmosphäre, sowie die Beschreibung ihres Verhaltens undder Vorhersagen, die aufgrund dessen getroen werden können. Viele andere Model-le zur mathematischen Beschreibung des Ozeans und der Atmosphäre basieren aufden primitiven Gleichungen, wodurch sie zur Grundlage sämtlicher Betrachtungen desOzeans und der Atmosphäre werden. Ziel dieser Arbeit war es, die Primitiven Gleichun-gen für den Ozean und die Atmosphäre aus den allgemeinen physikalischen Gesetzenherzuleiten, verschiedene Rand- und Anfangsbedingungen und einige Spezialfälle zubetrachten.

ii

Acknowledgments

I would like to thank my advisor, Peter Gangl for the time he has invested in workingwith me on this project. My gratitude also goes to professor Ulrich Langer for teachingme the necessary basics in this eld of study in his lecture and also for oering me alot of support.

A big word of thanks nally goes to my mum, who always had my back when I hadto focus on my studies and needed as little distraction as possible. Thanks for alwayssupporting me!

Michaela LehnerLinz, October 2017

iii

Contents

1 Introduction 1

2 Derivation of the Navier-Stokes Equations 3

2.1 Lagrangian and Eulerian Coordinates . . . . . . . . . . . . . . . . . . . 32.1.1 Lagrangian Coordinates . . . . . . . . . . . . . . . . . . . . . . 32.1.2 Eulerian Coordinates . . . . . . . . . . . . . . . . . . . . . . . . 3

2.2 Material Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.3 Reynolds' Transport Theorem . . . . . . . . . . . . . . . . . . . . . . . 42.4 The Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.4.1 Continuity Equation . . . . . . . . . . . . . . . . . . . . . . . . 62.4.2 Momentum Equation . . . . . . . . . . . . . . . . . . . . . . . . 62.4.3 Navier Stokes Equations . . . . . . . . . . . . . . . . . . . . . . 8

3 Derivation of the thermodynamic equations 9

3.1 Equation of State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.1.1 Ocean . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.1.2 Atmosphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

3.2 Thermodynamic Equations . . . . . . . . . . . . . . . . . . . . . . . . . 103.2.1 Atmosphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.2.2 Ocean . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

4 Description of physical eects 14

4.1 Forces on the momentum equation . . . . . . . . . . . . . . . . . . . . 144.1.1 Pressure Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144.1.2 Gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144.1.3 Coriolis Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154.1.4 Heat dissipation . . . . . . . . . . . . . . . . . . . . . . . . . . . 164.1.5 Momentum equation with physical eects . . . . . . . . . . . . . 16

4.2 Diusion of Salinity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

5 Derivation of the Primitive Equations 17

5.1 Spherical Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185.2 PEs of the Ocean . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

5.2.1 Boussinesq Approximation . . . . . . . . . . . . . . . . . . . . . 20

iv

CONTENTS

5.2.2 Hydrostatic Approximation . . . . . . . . . . . . . . . . . . . . 215.3 PEs of the Atmosphere . . . . . . . . . . . . . . . . . . . . . . . . . . . 235.4 Non-dimensional form of the PEs . . . . . . . . . . . . . . . . . . . . . 24

5.4.1 Ocean . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245.4.2 Atmosphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

6 Special Settings 27

6.1 PEs of the atmosphere in the p-coordinate system . . . . . . . . . . . . 276.2 Hydrostatic approximation with vertical velocity . . . . . . . . . . . . . 28

6.2.1 Ocean . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286.2.2 Atmosphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

7 Boundary Conditions for the PEs 29

7.1 Ocean . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297.2 Atmosphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

8 Conclusion 31

Bibliography 32

v

1 | Introduction

The goal of this thesis is to derive a set of equations capable of describing the physicalbehaviour of atmospheric and oceanic uids in a mathematical setting, while takinginto account the numeric stability and the mathematical complexity of this set of equa-tions.

When formulating models for the ocean and the atmosphere, we have to considersome physical properties rst. Since the ocean and the atmosphere have so many uid-dynamical properties in common, the study of one of them enriches our understandingof the other.

It is a well-established fact that the ocean can be seen as a slightly compressibleuid under Coriolis force. To describe the motions and states of the ocean, some basicquantities are necessary: the velocity eld, the temperature, the salinity, the pressureand the density of the seawater. We will have a closer look into the origin of thesequantities in Chapter 3.

The equations governing the behaviour of these quantities are the so-called generalequations of a compressible uid under Coriolis force, namely the momentum equa-tion, the continuity equation, the thermodynamic equation, the equation of state andthe equation of diusion of the salinity.

After collecting a set of general equations, we derive the primitive equations forthe ocean and the atmosphere. To do so, we consider the equations in spherical coor-dinates, since the earth can be approximated by a sphere. By applying the Boussinesqapproximation which neglects density dierences, and the hydrostatic approximation,which lets us ignore the z-coordinate, we can simplify our set of general equations toa set of primitive equations.

For special applications, especially observation of the long-term behaviour of theocean and the atmosphere, we cannot simply neglect the vertical velocity, as it is doneby the hydrostatic approximation. We will therefore also introduce a set of equationcalled the primitive equations with vertical velocity (PEV 2s), which are used espe-cially for this eld of application. For the atmosphere in particular, it is also useful toconsider the primitive equations in the so-called p-coordinate-system, which has some

1

CHAPTER 1. INTRODUCTION

advantages when it comes to calculations, reason being the continuity equation, whichcan in this coordinate setting be given in the form of an incompressible uid.

In Chapters 2-4, to establish a mathematical and physical foundation behind mod-els for the ocean and the atmosphere, we will derive the general equations of a com-pressible uid, to be more specic, the thermodynamic equation and the Navier Stokesequations, and describe the physical eects of the Coriolis force and the diusion ofsalinity. In Chapter 5, we will derive the so-called primitive equations, which will beused to model the ocean and the atmosphere. In Chapter 6, we will take into accountsome special settings and approximations. In Chapter 7, we will nally draw someconclusions.

2

2 | Derivation of theNavier-Stokes Equations

2.1 Lagrangian and Eulerian Coordinates

We want to describe the motion of a velocity eld during a time t under volume andsurface forces. There are two dierent ways to describe the movement of a uid (ow),namely the Eulerian and Lagrangian coordinates. While the Lagrangian frameworkdescribes a uid particle though its location at a time, the Eulerian formulation focuseson one specic point in the space through which the uid ows with a certain velocity,see e.g. [6].

2.1.1 Lagrangian Coordinates

Let (tS, tE) ⊂ (T1, T2) ⊂ R be our time interval for calculation and (T1, T2) be thetime interval where we observe the movement of a uid. Let Ω(t) ⊂ R3 be the that isoccupied by the uid at time t ∈ (T1, T2).If t0 ∈ (T1, T2) is a xed reference point, every uid particle in Ω(t0) can be identiedby its coordinates X = (X1, X2, X3) ∈ Ω(t0).The velocity of the uid in Lagrangian coordinates is then described by

v(X, t) =∂ϕ

∂t(X, t), (2.1)

where the vector function ϕ(X, t) is the trajectory of a specic uid particle.

2.1.2 Eulerian Coordinates

Euler describes the movement of a uid by its velocity eld. v(x, t) shall be the velocityof a uid particle in Eulerian coordinates, where we get with(x, t) ∈ (x, t) ∈ R4 : x = (x1, x2, x3) ∈ Ω(t), t ∈ (T1, T2):

v(x, t) = v(X, t) =∂ϕ

∂t(X, t), (2.2)

3

CHAPTER 2. DERIVATION OF THE NAVIER-STOKES EQUATIONS

where x = ϕ(X, t). It is more useful to use the Eulerian framework when handlinguids, since many physical factors are also given in Eulerian coordinates. Also, it iseasier to make comments about the behaviour of a uid when observing its velocityeld.

2.2 Material Derivative

Special care has to be taken when time-dependent quantities have to be transformedbetween the two frameworks. In other words, we are interested in the total timederivative of the property of a piece of uid. The total derivative is also known asmaterial derivative and is derived as follows. See e.g. [6] and [7].

Given a velocity eld v(x, t) and some scalar property f(x, t) for the uid, whichchanges in time, we get with the help of the chain rule, the following denition:

Denition 2.1 (Material derivative).

d

dtf(x, t) :=

∂

∂tf(x, t) +∇f(x, t) · v(x, t). (2.3)

2.3 Reynolds' Transport Theorem

Let ω(t) ∈ Ω(t) be an arbitrary, suciently smooth, simply connected domain, whichholds a xed amount of uid particles at time t ∈ (T1, T2):

ω(t) = x = ϕ(X, t) : X ∈ ω(t0) (2.4)

To deal with changes over time-dependent regions ω(t), a useful tool is the followingtheorem, as can also be seen in [6]:

Theorem 2.2 (Reynolds' Transport Theorem). Let t0 ∈ (T1, T2), ω(t0) be a bounded,suciently smooth domain with ω(t0) ⊂ Ω(t0). v : D → Rd and F : D → R withD :=

(x, t) ∈ Rd+1 : x ∈ Ω(t), t ∈ (T1, T2)

a C1 scalar function.

Then F :=∫ω(t)

F (x, t) dx is a well-dened C1 function with

dFdt

(t) =

∫ω(t)

[∂F

∂t(x, t) + div (F · v)(x, t)

]dx (2.5)

d

dt

∫ω(t)

F (x, t) dx =

∫ω(t)

[∂F

∂t(x, t) + div (F · v)(x, t)

]dx, (2.6)

where div(F · v) :=∑d

i=1∂(F ·vi)∂xi

=∑d

i=1∂F∂xivi + F · div v.

4


Proof. For simplicity, we give the proof of formula (2.5) for the one-dimensional caseonly. If we apply the substitution rule for d = 1, it holds

d

dt

∫ω(t)

F (x, t) dx =d

dt

∫ω(t0)

F (ϕ(X, t), t)∂ϕ

∂X(X, t) dX

=

∫ω(t0)

∂

∂t

[F (ϕ(X, t), t)

∂ϕ

∂X(X, t)

]dX

=

∫ω(t0)

(∂F

∂t(ϕ(X, t), t) +

∂F

∂ϕ(ϕ(X, t), t)

∂ϕ

∂t(X, t)

)∂ϕ

∂X(X, t) dX

+

∫ω(t0)

F (ϕ(X, t), t)∂

∂t

∂ϕ

∂X(X, t) dX

=

∫ω(t0)

(∂F

∂t(ϕ(X, t), t) +

∂F

∂ϕ(ϕ(X, t), t)v(X, t)

)∂ϕ

∂X(X, t) dX

+

∫ω(t0)

F (ϕ(X, t), t)∂

∂Xv(X, t) dX

=

∫ω(t)

(∂F

∂t(x, t) +

∂F

∂x(x, t)v(x, t)

)dx

+

∫ω(t)

F (x, t)∂

∂xv(x, t) dx

=

∫ω(t)

[∂F

∂t(x, t) +

∂F

∂x(x, t)v(x, t) + F (x, t)

∂v

∂x(x, t)

]dx

=

∫ω(t)

[∂F

∂t(x, t) +

∂

∂x(F · v)(x, t)

]dx

=

∫ω(t)

[∂F

∂t(x, t) + div (F · v)(x, t)

]dx.

For the three-dimensional case, the proof can be found e.g. in [12].

Remark 2.3. Applying Gauss' theorem, we can rewrite (2.5) in the following form:

d

dt

∫ω(t)

F (x, t) dx =

∫ω(t)

∂F

∂t(x, t) dx +

∫∂ω(t)

F (x, t)(v(x, t) · n(x, t))dsx (2.7)

(2.8)

5


2.4 The Equations of Motion

2.4.1 Continuity Equation

With the help of Axiom 2.4 about the conservation of mass and Reynolds' TransportTheorem 2.2, we can derive the so-called continuity equation of a compressible uid,which accounts for the ow of mass, see e.g. [6] and [7].

Axiom 2.4 (Conservation of mass). Let ω(t) = x = ϕ(X, t) : X ∈ ω(t0) be a regionwhich holds a xed amount of uid particles at time t ∈ (T1, T2). ω(t) will change itsform, but (if it's divergence-free), not its mass.

Since no mass can be created or destroyed, we get:

∂M∂t

(t) = 0 (2.9)

whereM(t) =∫ω(t)

ρ(x, t) dx, and ρ(x, t) is the density of the uid.

With Reynolds' transport theorem, we immediately get

0 =∂M∂t

(t) =∂

∂t

∫ω(t)

ρ(x, t) dx =

∫ω(t)

[∂ρ

∂t(x, t) + div (ρ · v)(x, t)

]dx (2.10)

for all t ∈ (T1, T2) and for all bounded and suciently smooth domains with ω(t) ⊂Ω(t0).

From this, it follows that

∂ρ

∂t(x, t) + div (ρ · v)(x, t) = 0 ∀(x, t) ∈ D (2.11)

which is nothing but the continuity equation.

By means of∑3

i=1∂(ρ·vi)∂xi

=∑3

i=1

[∂ρ∂xivi + ρ ∂vi

∂xi

], and with Denition 2.1 of the

material derivative dρdt

:= ∂ρ∂t

+∇ρ · v = ∂ρ∂t

+ ρ · ∇v, (2.11) is obviously equivalent to

d

dtρ(x, t) + ρ(x, t) div v(x, t) = 0 (2.12)

2.4.2 Momentum Equation

To describe the behaviour of the momentum of a uid responds to internal and imposedforces, we need the so-called momentum equation. See e.g. [4] for reference.Following from the second and third law of Newton, we get the following axiom:

6


Axiom 2.5 (Conservation of Momentum). Changes of the momentum in time in aclosed system of mass ω(t) = x = ϕ(X, t) : X ∈ ω(t0) in time equals the forcesacting on the system.

dI(t)

dt= F (ω(t)), (2.13)

where I(t) =∫ω(t)

v(x, t)ρ(x, t) dx.

We call F (ω(t)) = FB(ω(t) + FS(ω(t)) the forces acting on the system, which canbe split into

body forces on ω(t): FB(ω(t)) =

∫ω(t)

ρ(x, t)f(x, t) dx (2.14)

surface forces on ω(t): FS(ω(t)) =

∫∂ω(t)

t(n)(x, t)dsx (2.15)

where f(x, t) is some external force acting on the system, and t(n)(x, t) is the so-calledtotal strain in a point x ∈ ∂ω(t) at time t with normal vector n.

The so-called transformation formula gives us the relationship

tn(x,t)(x, t) =

[3∑j=1

σij(x, t)nj(x, t)

]i=1,3

(2.16)

between the stress tensor σ(x, t) and the total strain, as can also be seen in [6].

Applying Gauss' integration theorem to FS(ω(t)) provides

FS(ω(t)) =

∫∂ω(t)

t(n)(x, t)dsx

=

[∫∂ω(t)

3∑j=1

σij(x, t)nj(x, t)dsx

]i=1,3

=

[∫ω(t)

3∑j=1

∂σij∂xj

(x, t) dx

]i=1,3

=

∫ω(t)

div (σ(x, t)) dx

(2.17)

From Axiom 2.5 and Theorem 2.2, we get for i = 1, 3:

dIi(t)dt

=d

dt

∫ω(t)

vi(x, t)ρ(x, t) dx

=

∫ω(t)

∂(viρ)

∂t(x, t) + div (viρ · v)(x, t) dx.

(2.18)

7


Also, using our denition of the forces acting on a system ((2.14)-(2.15)), and theconsecutive calculations, we get

Fi(ω(t)) =

∫ω(t)

ρ(x, t)fi(x, t) dx +

∫ω(t)

3∑j=1

∂σij∂xj

(x, t) dx (2.19)

for i = 1, 3.

Those two equations are equal. In conclusion, we can write the equation of motionin conservative form. For i = 1, 3, it holds that:

∂ρvi∂t

(x, t) + div (ρvi · v)(x, t) =3∑j=1

∂σij∂xj

(x, t) + ρ(x, t)fi(x, t) (2.20)

Therefore, we get the equation of motion in convective form

∂ρ

∂tvi + ρ

∂vi∂t

+3∑j=1

[∂(ρvi)

∂xjvj + ρvi

∂vj∂xj

]=

3∑j=1

∂σij∂xj

(x, t) + ρfi, (2.21)

for i = 1, 3. Using the continuity equation (2.11), we can transform (2.21) into theform

ρ∂v

∂t+ ρv · ∇v = divσ + ρf (2.22)

on the whole domain D =

(x, t) ∈ Rd+1 : x ∈ Ω(t), t ∈ (T1, T2).

By using the denition of the material derivative (2.3), this becomes

ρdv

dt= divσ + ρf. (2.23)

See also [4] for reference.

2.4.3 Navier Stokes Equations

Together, the momentum equation and the continuity equation are called the NavierStokes Equations for a compressible uid:

ρdv

dt= divσ + ρf, (2.24)

dρ

dt+ ρdiv v = 0. (2.25)

We still have to dene the forces f in the momentum equation. This will be donein a separate chapter.

8

3 | Derivation of thethermodynamic equations

3.1 Equation of State

Since the momentum and the continuity equation provide us with four equations (themomentum equation is nothing but 3 coupled PDEs), but ve unknowns, it is necessaryto add another equation to our set of equations describing oceanic and atmosphericuids. The equation of state relates the various thermodynamic variables, tempera-ture, pressure, composition (salinity) and density to each other, see e.g [4].Generally speaking, the equation of state is the following:

ρ = f(T, S, p), (3.1)

where ρ is the density, T the temperature, S the salinity and p the pressure.

It is necessary for our models to provide an explicit equation of state.

3.1.1 Ocean

When thinking about the behaviour of the ocean, it is natural to expect ρ to decreaseif the temperature increases, and ρ to increase if the salinity increases. Therefore wecan formulate a linear law for the ocean, where ρ0, T0 and S0 are reference values ofdensity, temperature and salinity, and βT and βS are constant expansion coecients:

ρ = ρ0(1− βT (T − T0) + βS(S − S0)) (3.2)

3.1.2 Atmosphere

We can consider the atmosphere as an ideal gas. Since the state of a gas can bedetermined by its temperature, pressure and volume, by reformulating the density interms of pressure we get an equation of state in the form

p = RρT, (3.3)

where R = cp − cv is the specic gas constant, obtained by subtracting the specicheat at constant volume cv from the specic heat at constant pressure cp.

9

CHAPTER 3. DERIVATION OF THE THERMODYNAMIC EQUATIONS

3.2 Thermodynamic Equations

In uids where the equation of state involves temperature, the thermodynamic equa-tion is necessary for obtaining a closed system of equations. See e.g. [4]

We consider the ocean and the atmosphere as systems in equilibrium. Therefore, wecan express the internal energy I per unit mass of a system as function of the specicvolume α = 1/ρ, the specic entropy η and the chemical composition, parametrizedas the salinity S:

I = I(α, η, S), (3.4)

It should be noted that for the atmosphere, we will neglect the chemical composition.(3.4) is called the fundamental equation of state, for which the rst dierential

formally gives

dI =∂I

∂αdα +

∂I

∂ηdη +

∂I

∂SdS. (3.5)

Basis for our derivation of the thermodynamic equation is the rst law of ther-modynamics, which states the principle of conservation of energy, which again tellsus that the internal energy of a body might change due to work done, heat inputor maybe changes in its chemical composition, while the total amount of energy willalways stay the same.

Axiom 3.1 (First law of thermodynamics). The internal energy of an isolated systemis constant:

dI = dQ− dW + dC, (3.6)

where dW is the work done by the body, dQ is the heat input to the body, dC denotesthe changes in the chemical composition of the body (e.g. salinity), and dI is nallythe change in internal energy per unit mass.

The work, heat, and chemical composition can change due to various reasons. Wewant to specify them according to our requirements:

The second law of thermodynamics tells us that heat in a system can only ow inone direction (e.g. from a warmer to a colder location), but not backwards. We canalso formulate this in terms of entropy:

Axiom 3.2 (Second law of thermodynamics). Increase of the entropy times tempera-ture results from a transfer of heat:

T dη = dQ (3.7)

10


The work done by a body equals the pressure p times the change in volume α:

dW = p dα. (3.8)

Changes in chemical composition can in the case of the ocean be related to thechange in salinity dS times the chemical potential µ of the solution:

dC = µ dS (3.9)

In case of the atmosphere, we have of course no salinity, but changes in the amountof water in the air. These changes are brought about by changes in the temperature,which are already covered by the heat input. Therefore, for the atmosphere, we omitthe changes in chemical composition.

Adding up all we know about the conversation of energy, the equations (3.5)-(3.9)result in the fundamental thermodynamic relation

dI = T dη − p dα + µ dS. (3.10)

Assuming that locally the uid is in thermodynamic equilibrium. Then the ther-modynamic quantities (e.g. temperature, pressure, density,...) can vary in space, butlocally they are related by the equation of state and Maxwell's relations.

3.2.1 Atmosphere

As we said before, we omit changes in chemical composition for the atmosphere, re-sulting in the thermodynamic relation

dI = −p dα + dQ (3.11)

Since we can see the atmosphere as an ideal gas at constant pressure cp and constantvolume cv, the internal energy is a function of temperature only: dI = cv dT .

dQ = cv dT + p dα (3.12)

Using the relations α = RT/p and cp − cv = R, we get

dQ = cp dT − α dp (3.13)

By using the fact that, locally, the atmosphere is in thermodynamic equilibrium,we can form the material derivative:

dQ

dt= cp

dT

dt− RT

p

dp

dt(3.14)

where dQdt

is called the heat ux per unit density in a unit time interval.

11


3.2.2 Ocean

Reformulating the fundamental thermodynamic relation (3.10) in terms of entropyleaves us with the equation

dη =1

T( dI + p dα− µ dS)

=1

T(cv dT + p dα− µ dS)

=1

T( dQ− µ dS)

(3.15)

Which becomes, after forming the material derivative and with the help of (3.11), theso-called entropy equation:

dη

dt=

1

T

(dQ

dt− µdS

dt

)(3.16)

If we take the entropy to be a function of pressure, temperature and salinity, weget

T dη = T

(∂η

∂T

)p,S

dT + T

(∂η

∂p

)T,S

dp+ T

(∂η

∂S

)T,p

dS

= cp dT + T

(∂η

∂p

)T,S

dp+ T

(∂η

∂S

)T,p

dS

(3.17)

From equation (3.16) and (3.17), we get:

cpdT

dt+ T

(∂η

∂p

)T,S

dp

dt+ T

∂η

∂S

dS

dt=

dQ

dt− µdS

dt

dT

dt+T

cp

(∂η

∂p

)T,S

dp

dt+T

cp

∂η

∂S

dS

dt=

1

cp

dQ

dt− µ

cp

dS

dt

dT

dt+T

cp

(∂η

∂p

)T,S

dp

dt=

1

cp

dQ

dt− µ

cp

dS

dt− T

cp

∂η

∂S

dS

dt︸︷︷︸=:QT

(3.18)

where we will call QT the heat diusion.

For further calculations, we will need Maxwell's relations.

Theorem 3.3 (third of Maxwell's Relations).(∂η

∂p

)T

= −(∂α

∂T

)p

(3.19)

12


Proof. Due to (3.7) and (3.11), We can write

dI = T dη − p dα = d(T dη)− η dT − d(pα) + α dp,

dG = −η dT + α dp,

where G is the Gibbs function. Formally, we have now

dG =

(∂G

∂T

)p

dT +

(∂G

∂p

)T

dP.

From the last two equations, we see that η = −(∂G/∂T )p and α = (∂G/∂p)r. Since

∂2G

∂p∂T=∂2G

∂t∂p

it holds that (∂η

∂p

)T

= −(∂α

∂T

)p

Using this theorem, we get

dT

dt− T

cp

(∂α

∂T

)p

dp

dt= QT (3.20)

Since density and temperature can be related through a measurable coecient ofthermal expansion βT , we get: (

∂α

∂T

)p

=βTρ

(3.21)

With this, the thermodynamic equation gets the form

dT

dt− βTT

cpρ

dp

dt= QT (3.22)

Because liquids are characterized by a small thermal expansion coecient, it issometimes acceptable to neglect the second term on the left-hand side of equation(3.22),resulting in the thermodynamic equation for the ocean

dT

dt= QT (3.23)

Remark 3.4. The entropy equation and the internal energy equation are equivalentlyconnected via the equation of state. Both equations are usually referred to as thethermodynamic equation

All calculations in this chapter can also be found in [4].

13

4 | Description of physical eects

4.1 Forces on the momentum equation

The forces in the momentum equation consist of several physical factors, namely thepressure gradient, the gravity, the Coriolis force and the dissipative force. For furtherinformation, see [4] and [5]

4.1.1 Pressure Force

As stated in [4], when describing a uid, pressure is the normal force per unit areawithin or at the boundary of the uid. We call F the pressure force per unit value.

The pressure force on a domain of the uid is the integral of the pressure over itsboundary, in other words

Fp = −∫∂w(t)

p · n dsx (4.1)

The minus-sign comes from the direction of the pressure, which is pointed inwards,while the normal vector points outwards.

Applying the divergence theorem we get that

Fp = −∫w(t)

∇p dx (4.2)

The pressure force per unit volume can therefore be called the pressure gradient,dened as

F = −∇p (4.3)

4.1.2 Gravity

Gravity also has to be taken into account when dealing with atmospheric and oceanicuids. It is indicated by

g ≈ (0, 0,−9.8)[ms2

](4.4)

14

CHAPTER 4. DESCRIPTION OF PHYSICAL EFFECTS

4.1.3 Coriolis Force

Since the ocean and the atmosphere can be seen as layers of uid on a sphere, theirmotion is inuenced by rotation. We can include the eects of the rotation into ourequations by using the so-called Coriolis force, see e.g. [4]

Rate of change of a vector

To consider rotating uids, we have to relate the rate of change of a vector in theinertial and rotating frames. Following the steps in [4], we get a relation that we onlywant to dene here:

Denition 4.1 (Rate of change of a vector). Let B be the a vector that changes inthe inertial frame I. The rates of change in the inertial frame and the rotating frameR can be related by (

dB

dt

)I

=

(dB

dt

)R

+ Ω×B (4.5)

where we call Ω the angular velocity or rotation rate.

Remark 4.2. The rotation rate Ω is not to be confused with the domain Ω(t) we usedin Chapter 2 to dene the region occupied by the uid at time t.

Velocity in a rotating frame

By applying (4.5) on the position of a particle x = ϕ(X, t), we get the relation

vI = vR + Ω× x (4.6)

If we apply (4.5) in terms of vR on the above equation, we get(d

dt(vI − Ω× x)

)I

=

(dvRdt

)R

+ Ω× vR (4.7)(dvIdt

)I

=

(dvRdt

)R

+ Ω× vR +dΩ

dt× x + Ω× vI (4.8)

Under the assumption that the rotation rate Ω is constant, we get(dvRdt

)R

=

(dvIdt

)I

− 2Ω× vR − Ω× (Ω× x) (4.9)

where 2Ω × vR is the Coriolis force per unit mass and Ω × (Ω × x) is the centrifugalforce, which we will omit in our model. The Coriolis force is a so-called pseudo-forcewith basic properties:

• There is no Coriolis force on stationary bodies in the rotating frame

• The Coriols force deects moving bodies at right angle to their travel direction

• The Coriolis force does not work on a body due to its perpendicularity to thevelocity, therefore vR · (Ω× vR) = 0.

15

CHAPTER 4. DESCRIPTION OF PHYSICAL EFFECTS

4.1.4 Heat dissipation

The heat dissipation D is usually related to the stress tensor σ. See also [10] forreference.

D = −1

ρdivσ (4.10)

4.1.5 Momentum equation with physical eects

Applying pressure force, gravity, Coriolis force and heat dissipation on the momentumequation leaves us with the equation

ρdvRdt

+ 2ρΩ× vR +∇p+ ρg = D (4.11)

with vR being the velocity eld on the rotating frame. We will hereafter call thisvelocity eld in 3D V 3, to match with the references [1] and [2].

4.2 Diusion of Salinity

As given in [4], the diusion of salinity represents the eects of vaporation and precip-itation at the ocean surface, as well as molecular diusion. It changes only when thereare non-conservative sources and sinks, and is therefore determined by the conservationequation

dS

dt= QS (4.12)

where we call QS the salinity diusion.

16

5 | Derivation of the PrimitiveEquations

The goal of this chapter will be to derive the primitive equations (PEs) for the oceanand the atmosphere. We use the papers [1] and [2] as our main reference.

In the previous chapters, we derived equations describing atmospheric and oceanicuids mathematically. Together we call them the general equations of the ocean andthe atmosphere.

General equations of the ocean

To describe the ocean we need the momentum equation (4.11), the continuity equation(2.12), the thermodynamic equation (3.23), the equation of state (3.1) and the equationof diusion of the salinity (4.12):

ρdV 3

dt+ 2ρΩ× V 3 +∇p+ ρg = D (5.1)

dρ

dt+ ρ divV 3 = 0 (5.2)

dT

dt= QT (5.3)

dS

dt= QS (5.4)

ρ = f(T, S, p) (5.5)

where g is the gravity vector, D the molecular dissipation, QT and QS the heat andsalinity diusions. It is easily seen that we have now 7 equations for 7 unknowns.

17

CHAPTER 5. DERIVATION OF THE PRIMITIVE EQUATIONS

General equations of the atmosphere

For the atmosphere we can as well write down a set of general equations, containingthe momentum equation (4.11), the continuity equation (2.12), the thermodynamicequation (3.14) and the equation of state (3.3), or in other words - 6 equations for 6unknowns:

dV 3

dt= −1

ρ∇p− g − 2Ω× V 3 +D (5.6)

dρ

dt+ ρ divV 3 = 0 (5.7)

cpdT

dt− RT

p

dp

dt=

dQ

dt(5.8)

p = RρT (5.9)

5.1 Spherical Coordinates

Figure 5.1: sphericalcoordinates

Since the earth can be seen as a sphere, it makes sense toconsider our equations in spherical coordinates (θ, ϕ, r), where

0 ≤ θ ≤ π ... colatitude of the earth

0 ≤ ϕ ≤ 2π ... longitude of the earth

r ... radical distance

a ... radius of the earth

z = r − a ... vertical coordinate with respect to the sea level

as can be seen in Figure 5.11.

We know that the velocity of a uid is the time derivativeof a position. In terms of Cartesian coordinates, a position isdescribed as

x =

r cosϕ sin θr sinϕ sin θr cos θ

(5.10)

1Attribution: By Ag2gaeh (Own work) [CC BY-SA 4.0 (https://creativecommons.org/licenses/by-sa/4.0)], via Wikimedia Commons

18


The time derivative of this position can be found by

x =∂x

∂t=

r cosϕ sin θ − r sinϕ sin θϕ+ r cosϕ cos θθ

r sinϕ sin θ + r cosϕ sin θϕ+ r sinϕ cos θθ

r cos θ − r sin θθ

(5.11)

= r

cosϕ sin θsinϕ sin θ

cos θ

︸︷︷︸

ez

+r sin θϕ

− sinϕcosϕ

0

︸︷︷︸

eϕ

+rθ

cosϕ cos θsinϕ cos θ− sin θ

︸︷︷︸

eθ

(5.12)

where the dot always indicates the derivation with respect to time. See e.g. [9].

Velocity eld

In this setting, let eθ, eϕ, ez be the unit-vectors in θ-, ϕ- and z-directions respectively.Knowing this, we can write the velocity eld V 3 as the total of the horizontal velocityeld v = vθeθ + vϕeϕ and the vertical velocity eld w = vrez:

V 3 = v + w

= vθeθ + vϕeϕ + vrez (5.13)

= rθeθ + r sin θϕeϕ + rez (5.14)

Material Derivative

The material derivative in the spherical coordinate setting yields

d

dt=

∂

∂t+vθr

∂

∂θ+

vϕr sin θ

∂

∂ϕ+ vz

∂

∂z(5.15)

which we can see after some calculation.

Coriolis Force

The Coriolis force needs to be written in terms of the according unit vectors as can beseen in [5]:

Ω =

0 0 −1− sin θ cos θ 0cos θ sin θ 0

0Ω0

= (0,Ω cos θ,Ω sin θ) (5.16)

Hence:

2Ω× V 3 =

∣∣∣∣∣∣er eθ eϕ0 2Ω cos θ 2Ω sin θvr vθ vϕ

∣∣∣∣∣∣ (5.17)

= ez(2Ωvϕ cos θ − 2Ωvθ sin θ) + eθ2Ωvr sin θ − eϕ2Ωvr cos θ (5.18)

19


5.2 PEs of the Ocean

From theoretical and computational point of view, the general set of equations wefound for the ocean and the atmosphere are too complicated to study. Therefore, wewant to simplify this set of equations as much as possible, starting with the so-calledBoussinesq approximation, see e.g. [1]

5.2.1 Boussinesq Approximation

In the Boussinesq approximation, we neglect density dierences in the general equa-tions, with exception of the equation of state and the buoyancy term in the momentumequation. Therefore, the Navier Stokes equations (5.1) and (5.2) simplify to

ρ0dV 3

dt+ 2ρ0Ω× V 3 +∇p+ ρg = D (5.19)

divV 3 = 0 (5.20)

where ρ0 is some reference density. From the small depth assumption (the fact that thedepth of the ocean is small compared to the radius of the earth a) we get the argumentto further simplify the equations by replacing r by a to rst order. Particularly, thematerial derivative (2.3) becomes

d

dt=

∂

∂t+vθa

∂

∂θ+

vϕa sin θ

∂

∂ϕ+ vz

∂

∂z(5.21)

Taking into account the viscosity, our set of equations can be reformulated torepresent the Boussinesq equations of the ocean (BEs) in spherical coordinates, seee.g. [1]:

∂v

∂t+∇vv + w

∂v

∂z+

1

ρ0grad p+ 2Ω cos θk × v − µ∆v − ν ∂

2v

∂z2= 0 (5.22)

∂w

∂t+∇vw + w

∂w

∂z+

1

ρ0

∂p

∂z+

ρ

ρ0g − µ∆w − ν ∂

2w

∂z2= 0 (5.23)

div v +∂w

∂z= 0 (5.24)

∂T

∂t+∇vT + w

∂T

∂z− µT∆T − νT

∂2T

∂z2= 0 (5.25)

∂S

∂t+∇vS + w

∂S

∂z− µS∆S − νS

∂2S

∂z2= 0 (5.26)

ρ = ρ0(1− βT (T − T0) + βS(S − S0) (5.27)

where the horizontal gradient operator and the horizontal divergence operator are

20


dened as follows:

grad p =1

a

∂p

∂θeθ +

1

a sin θ

∂p

∂ϕeϕ (5.28)

div (vθeθ + vϕeϕ) =1

a sin θ

(∂(vθ sin θ)

∂θ+∂vϕ∂ϕ

)(5.29)

Also, as can be seen in [1], we can formulate the derivatives ∇vv for a vectorfunction v and ∇vT for a scalar function T in spherical coordinates as

∇vT =vθa

∂T

∂θ+

vϕa sin θ

∂T

∂ϕ(5.30)

∇vv =

vθa

∂vθ∂θ

+vϕ

a sin θ

∂vθ∂ϕ− vϕvϕ

acot θ

eθ, (5.31)

+

vϕa

∂vϕ∂θ

+vϕ

a sin θ

∂vϕ∂ϕ

+vθvϕa

cot θ

eϕ, (5.32)

and similarly, the Laplace-operators ∆ for scalar functions and vector elds become

∆T =1

a2 sin θ

∂

∂θ(sin θ

∂T

∂θ) +

1

sin θ

∂2T

∂ϕ2

, (5.33)

∆v = ∆ (vθeθ + vϕeϕ) (5.34)

=

∆vθ −

2 cos θ

a2 sin2 θ

∂vϕ∂ϕ− vθa2 sin2 θ

eθ (5.35)

+

∆vϕ +

2 cos θ

a2 sin2 θ

∂vθ∂ϕ− vϕa2 sin2 θ

eϕ. (5.36)

We also write the momentum equation (5.1) corresponding to the unit vectors ofthe velocity eld, resulting in a set of two equations describing the momentum in everydirection.

The Boussinesq equations are fully non-linear and 3-dimensional. Therefore, com-putational diculties will occur. However, with this set of equations, it is alreadypossible to get a mathematical analysis. In the next subsection, we will introduce thehydrostatic assumption, which is considered to be extremely accurate, and allows usto further simplify our set of equations, resulting in the primitive equations of the largescale ocean.

5.2.2 Hydrostatic Approximation

It is known that for the large scale ocean, the vertical scale is much smaller thanthe horizontal one. The scale analysis shows that the large-scale ocean satises thehydrostatic approximation:

∂p

∂z= −ρg (5.37)

21


This equation connects the pressure and the density. Due to its high accuracy, thishas become a fundamental equation in oceanography.

Our next step will be to replace the vertical momentum equation by the hydro-static approximation. Doing this, we drop the vertical velocity, which is not explicitlygiven in the hydrostatic equation. Mathematical justication is given in [11], and willnot be further discussed in this thesis.

As a result, we are not able to make predictions for w by doing so. We thereforehave to nd w through other means, which can cause some mathematical diculties.Solutions to overcome these diculties are given in [1].

The set of equations resulting from those approximations are called the primitiveequations of the large-scale ocean:

∂v

∂t+∇vv + w

∂v

∂z+

1


2v

∂z2= 0 (5.38)

∂p

∂z= −ρg (5.39)

div v +∂w

∂z= 0 (5.40)

∂T

∂t+∇vT + w

∂T


∂2T

∂z2= 0 (5.41)

∂S

∂t+∇vS + w

∂S


∂2S

∂z2= 0 (5.42)

ρ = ρ0(1− βT (T − T0) + βS(S − S0)) (5.43)

The hydrostatic equations make the PEs suitable for numerical computations,which makes them fundamental for oceanographic simulations. Other models of theocean can be derived from the PEs.

22


5.3 PEs of the Atmosphere

We get the primitive equations for the atmosphere the same way we got the primitiveequations for the ocean. As before, we will work with a spherical coordinate system.

Consequently, we can rewrite the general equations of the atmosphere as

dvθdt

+1

r(vrvθ − v2ϕ cot θ) = − 1

ρr

∂p

∂θ+ 2Ω cos θvϕ +Dθ (5.44)

dvϕdt

+1

r(vrvϕ + vθvϕ cot θ) = − 1

ρr sin θ

∂p

∂ϕ− 2Ω cos θvθ − 2Ω sin θvr +Dϕ (5.45)

dvrdt

+1

r(v2θ − v2ϕ) = −1

ρ

∂p

∂r− g + 2Ω sin θvϕ +Dr (5.46)

dρ

dt+ ρ

(1

r sin θ

∂vθ sin θ

∂θ+

1

r sin θ

∂vϕ∂ϕ

+1

r2∂r2vr∂r

)= 0 (5.47)

cpdT

dt− RT

p

dp

dt=

dQ

dt(5.48)

p = RρT (5.49)

Again, we will neglect density dierences as required for the Boussinesq approx-imation. Using this, we can rewrite the set of equations governing the atmosphereas

dvθdt−∂v2ϕ∂a

cot θ = − 1

ρa

∂p


dvϕdt−∂vθv

2ϕ

∂acot θ = − 1

ρa sin θ

∂p

∂ϕ+ 2Ω cosϕvθ +Dϕ (5.51)

dvrdt− 1

r(v2θ + v2ϕ) = −1

ρ

∂p

∂r− g + 2Ω sin θvϕ +Dr (5.52)

dρ

dt+ ρ

(1

a sin θ

∂vθ sin θ

∂θ+

1

a sin θ

∂vϕ∂ϕ

+∂vr∂r

)= 0 (5.53)

cpdT

dt− RT

p

dp

dt=

dQ

dt(5.54)

p = RρT (5.55)

Next, we use the hydrostatic approximation (5.37) connecting the pressure andthe density. Using all these approximations, we get the primitive equations for the

23


large-scale atmosphere

dvθdt−∂v2ϕ∂a

cot θ = − 1

ρa

∂p


dvϕdt−∂vθv

2ϕ

∂acot θ = − 1

ρa sin θ

∂p

∂ϕ+ 2Ω cosϕvθ +Dϕ (5.57)

∂p

∂r= −ρg (5.58)

dρ

dt+ ρ

(1

a sin θ

∂vθ sin θ

∂θ+

1

a sin θ

∂vϕ∂ϕ

+∂vr∂r

)= 0 (5.59)

cpdT

dt− RT

p

dp

dt=

dQ

dt(5.60)

p = RρT (5.61)

5.4 Non-dimensional form of the PEs

5.4.1 Ocean

Let U , T0, S0, ρ0 be reference values of the horizontal velocity, temperature, salinityand density respectively. Let a be the reference value for the horizontal length and Zthe reference value of the vertical length. Then we set

v = v′U w = w′εU T = T0T′

S = S0S′ ρ = ρ0ρ

′ p = ρ0U2p′

(5.62)

t =a

Ut′ h =

H

Zz = Zz′ = εaz′ (5.63)

f = 2 cos θ βT = βTT0 βS = βSS0

1

Re1=

µ

aU

1

Rt1=µTaU

1

Rs1=µSaU

1

Re2=

aν

Z2U

1

Rt2=

aνTZ2U

1

Rs2=

aνSZ2U

(5.64)

b =gZ

U2Ro =

U

aΩαT = ZαT τv =

ZτvU

Tb =TbT0

TA =TAT0

Sb =SbS0

ε =Z

a

(5.65)

where Rei are the Reynolds numbers, Rti and Rsi are the non-dimensional eddy-diusion coecients, Ro the Rossby number, which indicates how strongly the rota-tion of the earth inuences the dynamical behaviour of the ocean.

24


From there, we can compute the non-dimensional form of the primitive equationsof the ocean, resulting in:

∂v

∂t+∇v + w

∂v

∂z+ grad p+

1

Rofk × v − 1

Re1∆v − 1

Re2

∂2v

∂z2= 0 (5.66)

∂p

∂z+ bρ = 0 (5.67)

div v +∂w

∂z= 0 (5.68)

∂T

∂t+∇T + w

∂T

∂z− 1

Rt1∆T − 1

Rt2

∂2T

∂z2= 0 (5.69)

∂S

∂t+∇S + w

∂S

∂z− 1

Rs1∆S − 1

Rs2

∂2S

∂z2= 0 (5.70)

ρ = 1− βt(T − 1) + βS(S − 1) (5.71)

Remark 5.1. We simplied the equations by dropping the super index prime.

5.4.2 Atmosphere

Let, as before

v = v′U

w =P − p0a

Uw′

T = T0T′

Φ = U2Φ′

(5.72)

t =a

Ut′

p = (P − p0)ζ + p0

f ′ = 2 cos θ

(5.73)

1

Re1=

µ1

aU

1

Re2=

ν1ag2

UR2T 20

(P

P − p0

)2

1

Rt1=µ2T

20

aU3

1

Rt2=ν2ag

2

U3R2

(P

P − p0

)2

Ro =U

aΩ

a1 =R2T 2

0

C2U2b =

RT0(P − p0)U2P

αS = (P − p0)αS TS =TST0

(5.74)

25


As above, we have the same parameters for the Reynolds numbers and Rossbynumber, which in this case measures the inuence of the Earth's rotation on the be-haviour of the atmosphere.

Using these relations, we get the primitive equations for the ocean in non-dimensionalform (again, we omit the primes):

∂v

∂t+∇v + w

∂v

∂ζ+

f

Rok × v + gradΦ− 1

Re1∆v

− 1

Re2

∂

∂ζ

[(pT0PT

)2∂v

∂ζ

]= f1 (5.75)

div v +∂w

∂ζ= 0 (5.76)

∂Φ

∂ζ+bP

pT = 0 (5.77)

a1

(∂T

∂t+∇T + w

∂T

∂ζ

)− bP

pw − 1

Rt1∆T − 1

Rt2

∂

∂ζ

[(pT0PT

)2∂T

∂ζ

]= f2 (5.78)

26

6 | Special Settings

6.1 PEs of the atmosphere in the p-coordinate sys-

tem

From the hydrostatic approximation (5.37) it is easily seen that p is decreasing withrespect to the independent variable z, therefore we can perform a coordinate transfor-mation of the form

(t, θ, ϕ, z) −→ (t∗, θ∗, ϕ∗, p = p(t, θ, ϕ, z)) (6.1)

While z ∈ [0,∞), it holds that t ∈ [ps, 0] , namely from the pressure at sea or earthlevel to the pressure in the high atmosphere. We call this new system of variables thepressure coordinate system or p-coordinate system.

If we derive the primitive equations for the ocean from here with respect to thenew coordinates, we obtain after some work, see e.g [1].

dvθdt−v2ϕa

cot θ = −1

a

∂Φ

∂θ+ 2Ω cos θvϕ +Dθ, (6.2)

dvϕdt− vθvϕ

acot θ = − 1

a sin θ

∂Φ

∂ϕ+ 2Ω cos θvθ +Dϕ, (6.3)

∂Φ

∂p+R

pT = 0, (6.4)

∂w

∂p+

1

a sin θ

(∂vθ sin θ

∂θ+∂vϕ∂ϕ

)= 0, (6.5)

cp =dT

dt− RT

pw = QT , (6.6)

where Φ = gz is the geopotential.

We can observe that the continuity equation in the p-coordinate system takesthe same form as for an incompressible uid. It is known that the atmosphere is acompressible uid. Therefore, the form of the continuity equation is an advantage ofthe p-coordinate system.

27

CHAPTER 6. SPECIAL SETTINGS

6.2 Hydrostatic approximation with vertical viscos-

ity

6.2.1 Ocean

When studying the long-term behaviour of the ocean, viscosity will play an impor-tant role in the dynamics. A common method is therefore to replace the hydrostaticapproximation (5.37) by the following equation:

1

ρ0

∂p

∂z+

ρ

ρ0g − µ∆w − ν ∂

2w

∂z2= 0, (6.7)

which we will call the hydrostatic approximation with vertical viscosity.

By doing this, we get the primitive equations with vertical viscosity (PEV 2s):

∂v

∂t+∇vv + w

∂v

∂z+

1


2v

∂z2= 0 (6.8)

1

ρ0

∂p

∂z+

ρ

ρ0g − µ∆w − ν ∂

2w

∂z2= 0 (6.9)

div v +∂w

∂z= 0 (6.10)

∂T

∂t+∇vT + w

∂T


∂2T

∂z2= 0 (6.11)

∂S

∂t+∇vS + w

∂S


∂2S

∂z2= 0 (6.12)

ρ = ρ0(1− βT (T − T0) + βS(S − S0)) (6.13)

28

CHAPTER 6. SPECIAL SETTINGS

6.2.2 Atmosphere

Similarly, but with some computational eort, we can derive the PEV 2s with verticalviscosity for the atmosphere. See e.g. [2] for reference.

∂v

∂t+∇vv + w

∂v

∂ζ+

f

Rok × v + gradΦ− 1

Re1∆v − 1

Re2

∂

∂ζ

− 1

Re2

∂

∂ζ

[(pT0PT

)2∂v

∂ζ

]= f1 (6.14)

∂Φ

∂ζ+bP

pT − ε2

Re1∆w − ε2

Re2

∂

∂ζ

[(pT0PT

)2∂w

∂ζ

]= 0 (6.15)

div v +∂w

∂ζ= 0 (6.16)

a1

(∂T

∂t+∇vT + w

∂T

∂ζ

)− bP

pw − 1

Rt1∆T − 1

Rt2

∂

∂ζ

[(pT0PT

)2∂T

∂ζ

]= f2 (6.17)

29

7 | Boundary Conditions for the PEs

For a complete system of partial dierential equations it is of course necessary toinclude boundary and initial conditions. In the following section we formulate suitableboundary conditions.As for the initial conditions, they are necessary due to the derivations for time t in ourequations. Therefore we choose appropriate values for temperature, salinity, density,pressure and velocity in our model.

7.1 Ocean

Since we decided to approximate r by a, we can also approximate the domain lledby the sea by Ma ⊂ S2

a × R:

Ma =⋃

(θ,ϕ)∈Mah

(θ, ϕ) × (−H(θ, ϕ), 0) (7.1)

with S2a the sphere of radius a, Mah ⊂ S2

a a 2D-domain on the surface of the earth,which is occupied by the ocean, H(θ, ϕ) the depth of the ocean at the point of colati-tude (θ, ϕ)

When imagining the ocean, we see a large area of water with some islands andcontinents in it. When we remove those pieces of land, dened as I ia, (i = 1, 2, ..., N)from S2

a and only leave the water, we obtain the region Mah:

Mah = S2a \

(N⋃i=1

I ia

)

with the islands as simply connected open sets in S2a with smooth boundary such that

I ia ∩ Ija = ∅. for i 6= j

The depth function H : Mah → R is a smooth and positive function with H > 0in Mah, such that H is bounded from below on Mah: H ≥ Ho in Mah.

30

CHAPTER 7. BOUNDARY CONDITIONS FOR THE PES

With these assumptions and domains, we are able to give the boundary values forour equation system:

∂v∂z

= τvw = 0∂T∂z

= αT (TA − T )∂S∂z

= 0

Γu upper surface of the ocean, z = 0

(v, w) = 0(T, S) = (TH , SH)

Γb bottom of the ocean, z = −H

(v, w) = 0∂∂n

(T, S) = 0

Γl lateral boundary,

⋃(θ,ϕ)∈Mah

(θ, ϕ) × (H(θ, ϕ), 0)

where τv is the wind stress, αT a positive constant related to the turbulent heatingon the ocean surface, TA the atmospheric equilibrium temperature, TH and SH thetemperature and salinity at the bottom of the ocean.

7.2 Atmosphere

On the original coordinate system, we have no boundary conditions except some obvi-ous conditions resulting from periodicy. There are, however, boundary conditions onthe PEs if we write them in a p-coordinate-system, as can be seen in [2]

As before, we set p to be in the interval [p0, P ], where p0 > 0 is small and Papproximates the pressure at the Earth surface. In other words, p0 corresponds to theupper atmosphere and P to the surface of the Earth. The case where p0 = 0 is a bitcomplicated and requires special attention.

The boundary conditions for the primitive equations of the atmosphere in a p-coordinate system are

p = P : (v, w) = 0∂T

∂p= αS(TS − T ) (7.2)

p = p0 : (v, w) = 0∂T

∂p= 0 (7.3)

where αS = const and relates to the the turbulent transition on the Earth's surface,and TS is the given surface temperature.

For further insight, see [2].

31

8 | Conclusion

We have started from deriving and collecting the general physical equations of a com-pressible uid (under Coriolis force), to get a set of equations describing the oceanand the atmosphere. Under the aspect of physical eects, we were able to nd an evenbetter description for their behaviour.

However, since the general equations of the ocean and the atmosphere are not veryecient for mathematical purposes, we had to add further approximations in orderto simplify our model. With the help of the Boussinesq approximation and the hy-drostatic approximation we were able to simplify our set of equations and nd theprimitive equations of the ocean and the atmosphere, which are a very good mathe-matical approximation and are used to describe the behaviour of the two uids.

On further discussion, we created a non-dimensional form for the primitive equa-tions, and also took into account boundary conditions. For the atmosphere, it wasfurthermore possible to transform the coordinates to a p-coordinate system and get acontinuity equation in the same form as for a incompressible uid.

Another thing we looked into were the primitive equations with vertical viscosity,which are mainly used for discussions of the long-term behaviour of the ocean and theatmosphere.

32

Bibliography

[1] J.L. Lions, R. Temam, S. Wang On the equations of the large-scale ocean. Non-linearity, 5:1007-1053 1992

[2] J.L. Lions, R. Temam, S. Wang New formulations of the primitive equations ofatmosphere and applications. Nonlinearity, 5:237-288, 1992

[3] C. Hu, R. Temam, M. Ziane The primitive equations on the large scale oceanunder the small depth hypothesis. Discr. Cont. Dynam. Sys., 9(1):97-131, 2003

[4] G.K. Vallis Atmospheric and Oceanic Fluid Dynamics. Cambridge UniversityPress, 2006

[5] P. Berlo Lectures on Introduction to Geophysical Fluid Dynamics. Imperial Col-lege London.

[6] U. Langer Lecture on Mathematische Methoden in der Technik. Johannes KeplerUniversity, 2015

[7] S. Kindermann Lecture on an introductiont to mathematical methods for contin-uum mechanics. Johannes Kepler University, 2014

[8] J. Pedlosky Geophysical Fluid Dynamics. Springer New York, 1986.

[9] Mathworld Spherical Coordinates

[10] R. Bannister Primitive Equations: http://www.met.reading.ac.uk/ ross/Science/PrimEqs.html. University of Reading, 2000

[11] P. Azérad, F. Gullién Mathematical justication of the hydrostatic approximationin the primitive equations of geophysical uid dynamics. SIAM J. Math. Anal.,33(4), 847-859, 2001

[12] M. Feistauer Mathematical Methods in Fluid Dynamics. LongmanScientic &Technical, 1993

33

Eidesstattliche Erklärung

Ich, Michaela Lehner, erkläre an Eides statt, dass ich die vorliegende Bachelorarbeitselbständig und ohne fremde Hilfe verfasst, andere als die angegebenen Quellen undHilfsmittel nicht benutzt bzw. die wörtlich oder sinngemäÿ entnommenen Stellen alssolche kenntlich gemacht habe.

Linz, Oktober 2017

Michaela Lehner

Curriculum Vitae

Name: Michaela Lehner

Nationality: Austria

Date of Birth: 12 December, 1993

Place of Birth: Linz, Austria

Education:

19992003 Volksschule (elementary school)Ansfelden

20032012 Bundesrealgymnasium (secondary comprehensive school)Linz, Hamerlingstraÿe

20122017 Studies in Technical Mathematics,Johannes Kepler University Linz

oceanic and atmospheric fluid dynamics · with me on this project. my gratitude also goes to...

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