Dynamics of the atmosphere: a course of theoretical meteorology

DYNAMICS OF THE ATMOSPHERE:A COURSE IN THEORETICAL

METEOROLOGY

WILFORD ZDUNKOWSKIand

ANDREAS BOTT

published by the press syndicate of the university of cambridgeThe Pitt Building, Trumpington Street, Cambridge, United Kingdom

cambridge university pressThe Edinburgh Building, Cambridge CB2 2RU, UK40 West 20th Street, New York, NY 10011-4211, USA

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C© Cambridge University Press 2003

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no reproduction of any part may take place withoutthe written permission of Cambridge University Press.

First published 2003

Printed in the United Kingdom at the University Press, Cambridge

TypefaceTimes 11/14 pt SystemLATEX2ε [tb]

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Contents

Preface pagexvPart 1 Mathematical tools 1

M1 Algebra of vectors 3M1.1 Basic concepts and definitions 3M1.2 Reference frames 6M1.3 Vector multiplication 7M1.4 Reciprocal coordinate systems 15M1.5 Vector representations 19M1.6 Products of vectors in general coordinate systems 22M1.7 Problems 23

M2 Vector functions 25M2.1 Basic definitions and operations 25M2.2 Special dyadics 28M2.3 Principal-axis transformation of symmetric tensors 32M2.4 Invariants of a dyadic 34M2.5 Tensor algebra 40M2.6 Problems 42

M3 Differential relations 43M3.1 Differentiation of extensive functions 43M3.2 The Hamilton operator in generalized coordinate

systems 48M3.3 The spatial derivative of the basis vectors 51M3.4 Differential invariants in generalized coordinate systems 53M3.5 Additional applications 56M3.6 Problems 60

M4 Coordinate transformations 62M4.1 Transformation relations of time-independent

coordinate systems 62

vii

viii Contents

M4.2 Transformation relations of time-dependentcoordinate systems 67

M4.3 Problems 73M5 The method of covariant differentiation 75

M5.1 Spatial differentiation of vectors and dyadics 75M5.2 Time differentiation of vectors and dyadics 79M5.3 The local dyadic of vP 82M5.4 Problems 83

M6 Integral operations 84M6.1 Curves, surfaces, and volumes in the generalqi system 84M6.2 Line integrals, surface integrals, and volume integrals 87M6.3 Integral theorems 90M6.4 Fluid lines, surfaces, and volumes 94M6.5 Time differentiation of fluid integrals 96M6.6 The general form of the budget equation 101M6.7 Gauss’ theorem and the Dirac delta function 104M6.8 Solution of Poisson’s differential equation 106M6.9 Appendix: Remarks on Euclidian and Riemannian

spaces 107M6.10 Problems 110

M7 Introduction to the concepts of nonlinear dynamics 111M7.1 One-dimensional flow 111M7.2 Two-dimensional flow 116

Part 2 Dynamics of the atmosphere 1311 The laws of atmospheric motion 133

1.1 The equation of absolute motion 1331.2 The energy budget in the absolute reference system 1361.3 The geographical coordinate system 1371.4 The equation of relative motion 1461.5 The energy budget of the general relative system 1471.6 The decomposition of the equation of motion 1501.7 Problems 154

2 Scale analysis 1572.1 An outline of the method 1572.2 Practical formulation of the dimensionless flow

numbers 1592.3 Scale analysis of large-scale frictionless motion 1612.4 The geostrophic wind and the Euler wind 1672.5 The equation of motion on a tangential plane 1692.6 Problems 169

Contents ix

3 The material and the local description of flow 1713.1 The description of Lagrange 1713.2 Lagrange’s version of the continuity equation 1733.3 An example of the use of Lagrangian coordinates 1753.4 The local description of Euler 1823.5 Transformation from the Eulerian to the Lagrangian

system 1863.6 Problems 187

4 Atmospheric flow fields 1894.1 The velocity dyadic 1894.2 The deformation of the continuum 1934.3 Individual changes with time of geometric fluid

configurations 1994.4 Problems 205

5 The Navier–Stokes stress tensor 2065.1 The general stress tensor 2065.2 Equilibrium conditions in the stress field 2085.3 Symmetry of the stress tensor 2095.4 The frictional stress tensor and the deformation

dyadic 2105.5 Problems 212

6 The Helmholtz theorem 2146.1 The three-dimensional Helmholtz theorem 2146.2 The two-dimensional Helmholtz theorem 2166.3 Problems 217

7 Kinematics of two-dimensional flow 2187.1 Atmospheric flow fields 2187.2 Two-dimensional streamlines and normals 2227.3 Streamlines in a drifting coordinate system 2257.4 Problems 228

8 Natural coordinates 2308.1 Introduction 2308.2 Differential definitions of the coordinate lines 2328.3 Metric relationships 2358.4 Blaton’s equation 2368.5 Individual and local time derivatives of the velocity 2388.6 Differential invariants 2398.7 The equation of motion for frictionless horizontal flow 2428.8 The gradient wind relation 2438.9 Problems 244

x Contents

9 Boundary surfaces and boundary conditions 2469.1 Introduction 2469.2 Differential operations at discontinuity surfaces 2479.3 Particle invariance at boundary surfaces, displacement

velocities 2519.4 The kinematic boundary-surface condition 2539.5 The dynamic boundary-surface condition 2589.6 The zeroth-order discontinuity surface 2599.7 An example of a first-order discontinuity surface 2659.8 Problems 267

10 Circulation and vorticity theorems 26810.1 Ertel’s form of the continuity equation 26810.2 The baroclinic Weber transformation 27110.3 The baroclinic Ertel–Rossby invariant 27510.4 Circulation and vorticity theorems for frictionless

baroclinic flow 27610.5 Circulation and vorticity theorems for frictionless

barotropic flow 29310.6 Problems 301

11 Turbulent systems 30211.1 Simple averages and fluctuations 30211.2 Weighted averages and fluctuations 30411.3 Averaging the individual time derivative and the

budget operator 30611.4 Integral means 30711.5 Budget equations of the turbulent system 31011.6 The energy budget of the turbulent system 31311.7 Diagnostic and prognostic equations of turbulent

systems 31511.8 Production of entropy in the microturbulent system 31911.9 Problems 324

12 An excursion into spectral turbulence theory 32612.1 Fourier Representation of the continuity equation and

the equation of motion 32612.2 The budget equation for the amplitude of the

kinetic energy 33112.3 Isotropic conditions, the transition to the continuous

wavenumber space 33312.4 The Heisenberg spectrum 33612.5 Relations for the Heisenberg exchange coefficient 34012.6 A prognostic equation for the exchange coefficient 341

Contents xi

12.7 Concluding remarks on closure procedures 34612.8 Problems 348

13 The atmospheric boundary layer 34913.1 Introduction 34913.2 Prandtl-layer theory 35013.3 The Monin–Obukhov similarity theory of the neutral

Prandtl layer 35813.4 The Monin–Obukhov similarity theory of the diabatic

Prandtl layer 36213.5 Application of the Prandtl-layer theory in numerical

prognostic models 36913.6 The fluxes, the dissipation of energy, and the exchange

coefficients 37113.7 The interface condition at the earth’s surface 37213.8 The Ekman layer – the classical approach 37513.9 The composite Ekman layer 38113.10 Ekman pumping 38813.11 Appendix A: Dimensional analysis 39113.12 Appendix B: The mixing length 39413.13 Problems 396

14 Wave motion in the atmosphere 39814.1 The representation of waves 39814.2 The group velocity 40114.3 Perturbation theory 40314.4 Pure sound waves 40714.5 Sound waves and gravity waves 41014.6 Lamb waves 41814.7 Lee waves 41814.8 Propagation of energy 41814.9 External gravity waves 42214.10 Internal gravity waves 42614.11 Nonlinear waves in the atmosphere 43114.12 Problems 434

15 The barotropic model 43515.1 The basic assumptions of the barotropic model 43515.2 The unfiltered barotropic prediction model 43715.3 The filtered barotropic model 45015.4 Barotropic instability 45215.5 The mechanism of barotropic development 46315.6 Appendix 46815.7 Problems 470

xii Contents

16 Rossby waves 47116.1 One- and two-dimensional Rossby waves 47116.2 Three-dimensional Rossby waves 47616.3 Normal-mode considerations 47916.4 Energy transport by Rossby waves 48216.5 The influence of friction on the stationary Rossby wave 48316.6 Barotropic equatorial waves 48416.7 The principle of geostrophic adjustment 48716.8 Appendix 49316.9 Problems 494

17 Inertial and dynamic stability 49517.1 Inertial motion in a horizontally homogeneous

pressure field 49517.2 Inertial motion in a homogeneous geostrophic wind field 49717.3 Inertial motion in a geostrophic shear wind field 49817.4 Derivation of the stability criteria in the geostrophic

wind field 50117.5 Sectorial stability and instability 50417.6 Sectorial stability for normal atmospheric conditions 50917.7 Sectorial stability and instability with permanent

adaptation 51017.8 Problems 512

18 The equation of motion in general coordinate systems 51318.1 Introduction 51318.2 The covariant equation of motion in general coordinate

systems 51418.3 The contravariant equation of motion in general

coordinate systems 51818.4 The equation of motion in orthogonal coordinate systems 52018.5 Lagrange’s equation of motion 52318.6 Hamilton’s equation of motion 52718.7 Appendix 53018.8 Problems 531

19 The geographical coordinate system 53219.1 The equation of motion 53219.2 Application of Lagrange’s equation of motion 53619.3 The first metric simplification 53819.4 The coordinate simplification 53919.5 The continuity equation 54019.6 Problems 541

Contents xiii

20 The stereographic coordinate system 54220.1 The stereographic projection 54220.2 Metric forms in stereographic coordinates 54620.3 The absolute kinetic energy in stereographic coordinates 54920.4 The equation of motion in the stereographic

Cartesian coordinates 55020.5 The equation of motion in stereographic

cylindrical coordinates 55420.6 The continuity equation 55620.7 The equation of motion on the tangential plane 55820.8 The equation of motion in Lagrangian enumereation

coordinates 55920.9 Problems 564

21 Orography-following coordinate systems 56521.1 The metric of theη system 56521.2 The equation of motion in theη system 56821.3 The continuity equation in theη system 57121.4 Problems 571

22 The stereographic system with a generalized vertical coordinate 57222.1 Theξ transformation and resulting equations 57322.2 The pressure system 57722.3 The solution scheme using the pressure system 57922.4 The solution to a simplified prediction problem 58222.5 The solution scheme with a normalized pressure

coordinate 58422.6 The solution scheme with potential temperature as

vertical coordinate 58722.7 Problems 589

23 A quasi-geostrophic baroclinic model 59123.1 Introduction 59123.2 The first law of thermodynamics in various forms 59223.4 The vorticity and the divergence equation 59323.5 The first and second filter conditions 59523.6 The geostrophic approximation of the heat equation 59723.7 The geostrophic approximation of the vorticity equation 60323.8 Theω equation 60523.9 The Philipps approximation of the ageostrophic

component of the horizontal wind 60923.10 Applications of the Philipps wind 61423.11 Problems 617

xiv Contents

24 A two-level prognostic model, baroclinic instability 61924.1 Introduction 61924.2 The mathematical development of the two-level model 61924.3 The Phillips quasi-geostrophic two-level circulation model 62324.4 Baroclinic instability 62424.5 Problems 633

25 An excursion concerning numerical procedures 63425.1 Numerical stability of the one-dimensional

advection equation 63425.2 Application of forward-in-time and central-in-space

difference quotients 64025.3 A practical method for the elimination of the weak

instability 64225.4 The implicit method 64225.5 The aliasing error and nonlinear instability 64525.6 Problems 648

26 Modeling of atmospheric flow by spectral techniques 64926.1 Introduction 64926.2 The basic equations 65026.3 Horizontal discretization 65526.4 Problems 667

27 Predictability 66927.1 Derivation and discussion of the Lorenz equations 66927.2 The effect of uncertainties in the initial conditions 68127.3 Limitations of deterministic predictability of the

atmosphere 68327.4 Basic equations of the approximate stochastic

dynamic method 68927.5 Problems 692Answers to Problems 693List of frequently used symbols 702References and bibliography 706

M1

Algebra of vectors

M1.1 Basic concepts and definitions

A scalar is a quantity that is specified by its sign and by its magnitude. Examplesare temperature, the specific volume, and the humidity of the air. Scalars willbe written using Latin or Greek letters such asa, b, . . ., A,B, . . ., α, β, . . .. Avectorrequires for its complete characterization the specification of magnitude anddirection. Examples are the velocity vector and the force vector. A vector will berepresented by a boldfaced letter such asa,b, . . ., A,B, . . .. A unit vector is avector of prescribed direction and of magnitude 1. Employing the unit vectoreA,the arbitrary vectorA can be written as

A = |A|eA = AeA =⇒ eA = A|A| (M1.1)

Two vectorsA andB are equal if they have the same magnitude and directionregardless of the position of their initial points,

that is |A| = |B| andeA = eB . Two vectors arecollinear if they are parallelor antiparallel. Three vectors that lie in the same plane are calledcoplanar. Twovectors always lie in the same plane since they define the plane. The followingrules are valid:

the commutative law: A ± B = B± A, Aα = αA

the associative law: A + (B+ C) = (A + B) +C, α(βA) = (αβ)A

the distributive law: (α + β)A = αA + βA(M1.2)

The concept of linear dependence of a set of vectorsa1,a2, . . ., aN is closelyconnected with the dimensionality of space. The following definition applies: Aset ofN vectorsa1,a2, . . ., aN of the same dimension is linearly dependent if thereexists a set of numbersα1, α2, . . ., αN , not all of which are zero, such that

α1a1 + α2a2 + · · · + αNaN = 0 (M1.3)

3

4 Algebra of vectors

Fig. M1.1 Linear vector spaces: (a) one-dimensional, (b) two-dimensional, and (c) three-dimensional.

If no such numbers exist, the vectorsa1,a2, . . ., aN are said to be linearly inde-pendent. To get the geometric meaning of this definition, we consider the vectorsa andb as shown in Figure M1.1(a). We can find a numberk �= 0 such that

b = ka (M1.4a)

By settingk = −α/β we obtain the symmetrized form

αa+ βb = 0 (M1.4b)

Assuming that neitherα nor β is equal to zero then it follows from the abovedefinition that two collinear vectors are linearly dependent. They define the one-dimensionallinear vector space. Consider two noncollinear vectorsa andb asshown in Figure M1.1(b). Every vectorc in their plane can be represented by

c= k1a+ k2b or αa+ βb+ γ c= 0 (M1.5)

with a suitable choice of the constantsk1 andk2. Equation (M1.5) defines a two-dimensional linear vector space. Since not all constantsα, β, γ are zero, thisformula insures that the three vectors in the two-dimensional space are linearlydependent. Taking three noncoplanar vectorsa,b, andc, we can represent everyvectord in the form

d = k1a+ k2b+ k3c (M1.6)

in a three-dimensional linear vector space, see Figure M1.1(c). This can be gener-alized by stating that, in anN -dimensional linear vector space, every vector can berepresented in the form

x = k1a1 + k2a2 + · · · + kNaN (M1.7)

where thea1,a2, . . ., aN are linearly independent vectors. Any set of vectors con-taining more thanN vectors in this space is linearly dependent.

M1.1 Basic concepts and definitions 5

Table M1.1.Extensive quantities of different degrees for theN-dimensional linear vector space

Extensive Number of Number ofquantity Degreev Symbol vectors components

Scalar 0 B 0 N0 = 1Vector 1 B 1 N1

Dyadic 2 B 2 N2

Fig. M1.2 Projection of a vectorB onto a vectorA.

We call the set ofN linearly independent vectorsa1,a2, . . ., aN thebasis vectorsof theN -dimensional linear vector space. The numbersk1, k2, . . ., kN appearing in(M1.7) are themeasure numbersassociated with the basis vectors. The termkiaiof the vectorx in (M1.7) is thecomponent of this vectorin the directionai.

A vectorB may be projected onto the vectorA parallel to the direction of astraight linek as shown in Figure M1.2(a). If the direction of the straight linek isnot given, we perform an orthogonal projection as shown in part (b) of this figure.A projection in three-dimensional space requires a planeF parallel to which theprojection of the vectorB onto the vectorA can be carried out; see Figure M1.2(c).

In vector analysis anextensive quantityof degreeν is defined as a homogeneoussum of general products of vectors (with no dot or cross between the vectors). Thenumber of vectors in a product determines the degree of the extensive quantity.This definition may seem strange to begin with, but it will be familiar soon. Thus, ascalar is anextensivequantity of degreezero, anda vector is anextensivequantity ofdegree one. An extensive quantity of degree two is called adyadic. Every dyadicBmay be represented as the sumof three or moredyads.B = p1P1 + p2P2 + p3P3 +· · ·. Either theantecedentspi or theconsequentsPi may be arbitrarily assignedas long as they are linearly independent. Our practical work will be restricted toextensive quantities of degree two or less. Extensive quantities of degree three andfour also appear in the highly specialized literature. Table M1.1 gives a list ofextensive quantities used in our work. Thus, in the three-dimensional linear vectorspace withN = 3, a vector consists of three and a dyadic of nine components.


Fig. M1.3 The general vector basisq1,q2,q3 of the three-dimensional space.

M1.2 Reference frames

The representation of a vector in component form depends on the choice of aparticular coordinate system. Ageneral vector basisat a given point in three-dimensional space is defined by three arbitrary linearly independent basis vectorsq1,q2,q3 spanning the space. In general, the basis vectors are neither orthogonalnor unit vectors; they may also vary in space and in time.

Consider a position vectorr extending from an arbitrary origin to a pointP inspace. An arbitrary vectorA extending fromP is defined by the three basis vectorsqi , i = 1,2,3, existing atP at time t, as shown in Figure M1.3 for an obliquecoordinate system. Hence, the vectorA may be written as

A = A1q1 + A2q2 +A3q3 =3∑k=1

Akqk (M1.8)

where it should be observed that the so-calledaffine measure numbersA1, A2, A3

carry superscripts, and the basis vectorsq1,q2,q3 carry subscripts. This type of no-tation is used in theRicci calculus, which is the tensor calculus for nonorthonormalcoordinate systems. Furthermore, it should be noted that there must be an equalnumber of upper and lower indices.

Formula (M1.8) can bewritten more briefly with the help of the familiarEinsteinsummation conventionwhich omits the summation sign:

A = A1q1 +A2q2 +A3q3 = Anqn (M1.9)

We will agree on the following notation: Whenever an index (subscript or super-script)m,n, p, q, r, s, t , is repeated in a term, we are to sum over that index from 1to 3, ormore generally toN . In contrast to the summation indicesm,n, p, q, r, s, t,the lettersi, j, k, l are considered to be “free” indices that are used to enumerateequations. Note that summation is not implied even if the free indices occur twicein a term or even more often.

M1.3 Vector multiplication 7

Aspecial caseof thegeneral vector basis is theCartesianvector basisrepresentedby the three orthogonal unit vectorsi, j , k, or, more conveniently,i1, i2, i3. Each ofthese three unit vectors has the same direction at all points of space. However, inrotating coordinate systems these unit vectors also depend on time. The arbitraryvectorA may be represented by

A = Ax i + Ay j +Azk = Anin = Aninwith Ax = A1 = A1, Ay = A2 = A2, Az = A3 = A3

(M1.10)

In the Cartesian coordinate space there is no need to distinguish between upper andlower indices so that (M1.10) may be written in different ways. We will return tothis point later.

Finally, we wish to define theposition vectorr . In a Cartesian coordinate systemwe may simply write

r = xi + yj + zk = xnin = xnin (M1.11)

In an oblique coordinate system, provided that the same basis exists everywhere inspace, we may write the general form

r = q1q1 + q2q2 + q3q3 = qnqn (M1.12)

where theqi are the measure numbers corresponding to the basis vectorsqi. Theform (M1.12) is also valid along the radius in a spherical coordinate system sincethe basis vectors do not change along this direction.

A different situation arises in case of curvilinear coordinate lines since theorientations of the basis vectors changewith position. This is evident, for example,on considering the coordinate lines (lines of equal latitude and longitude) on thesurface of a nonrotating sphere. In case of curvilinear coordinate lines the positionvectorr has to be replaced by the differential expressiondr = dqn qn. Later wewill discuss this topic in the required detail.

M1.3 Vector multiplication

M1.3.1 The scalar product of two vectors

By definition, the coordinate-free form of the scalar product is given by

A · B = |A| |B| cos(A,B) (M1.13)


Fig. M1.4 Geometric interpretation of thescalar product.

If the vectorsA andB are orthogonal the expression cos(A,B) = 0 so that thescalar product vanishes. The following rules involving the scalar product are valid:

the commutative law: A · B = B · Athe associative law: (kA) · B = k(A · B) = kA · Bthe distributive law: A · (B+ C) = A · B+ A · C

(M1.14)

Moreover, we recognize that the scalar product, also known as the dot product orinner product, may be represented by the orthogonal projections

A · B = |A ′||B|, A · B = |A||B′| (M1.15)

whereby the vectorA ′ is the projection ofA onB, andB′ is the projection ofB onA; see Figure M1.4.

The component notation of the scalar product yields

A · B = A1B1q1· q1 + A1B2q1· q2 +A1B3q1· q3+A2B1q2· q1 +A2B2q2· q2 +A2B3q2· q3+A3B1q3· q1 +A3B2q3· q2 +A3B3q3· q3

(M1.16)

Thus, in general the scalar product results in nine terms. Utilizing the Einsteinsummation convention we obtain the compact notation

A · B = Amqm · Bnqn = AmBnqm · qn = AmBngmn (M1.17)

The quantitygij is knownas the covariantmetric fundamental quantityrepresentingan element of a covariant tensor of rank two or order two. This tensor is calledthemetric tensoror the fundamental tensor. The expression “covariant” will bedescribed later. Sinceqi ·qj = qj ·qi we have the identity

gij = gji (M1.18)


On substituting forA,B the unit vectors of the Cartesian coordinate system, wefind the well-known orthogonality conditions for the Cartesian unit vectors

i · j = 0, i · k = 0, j · k = 0 (M1.19)

or the normalization conditions

i · i = 1, j · j = 1, k · k = 1 (M1.20)

For the special case of Cartesian coordinates, from (M1.16) we, therefore, obtainfor the scalar product

A · B = AxBx +AyBy +AzBz (M1.21)

When thebasis vectorsi, j , k areorientedalong the (x, y, z)-axes, the coordinatesof their terminal points are given by

i : (1,0,0), j : (0,1,0), k : (0,0,1) (M1.22)

This expression is the Euclidian three-dimensional space or the space of ordinaryhuman life. On generalizing to theN -dimensional space we obtain

e1: (1,0, . . .,0), e2: (0,1, . . .,0), . . . eN: (0,0, . . .,1)

(M1.23)

This equation is known as the Cartesian reference frame of theN -dimensionalEuclidian space. In this space the generalized form of the position vectorr is givenby

r = x1e1 + x2e2 + · · · + xNeN (M1.24)

The length or the magnitude of the vectorr is also known as theEuclidian norm

|r | = √r ·r =

√(x1)2 + (x2)2 + · · · + (xN )2 (M1.25)

M1.3.2 The vector product of two vectors

In coordinate-free or invariant notation the vector product of two vectors is definedby

A × B = C = |A| |B| sin(A,B) eC (M1.26)

The unit vectoreC is perpendicular to the plane defined by the vectorsA andB.The direction of the vectorC is defined in such a way that the vectorsA, B, andC


Fig. M1.5 Geometric interpretation of the vector or cross product.

form a right-handed system. Themagnitude ofC is equal to the areaF of a parallel-ogram defined by the vectorsA andB as shown in Figure M1.5. Interchanging thevectorsA andB givesA × B = −B× A. This follows immediately from (M1.26)since the unit vectoreC now points in the opposite direction.

The following vector statements are valid:

A × (B+ C) = A × B+ A × C

(kA) × B = A × (kB) = kA × B

A × B = −B× A

(M1.27)

The component representation of the vector product yields

A × B = Amqm × Bnqn =

∣∣∣∣∣∣∣∣∣

q2 × q3 q3 × q1 q1 × q2A1 A2 A3

B1 B2 B3

∣∣∣∣∣∣∣∣∣(M1.28)

By utilizing Cartesian coordinates we obtain the well-known relation

A × B =

∣∣∣∣∣∣∣∣∣

i j k

Ax Ay Az

Bx By Bz

∣∣∣∣∣∣∣∣∣(M1.29)

M1.3.3 The dyadic representation, the general product of two vectors

The general ordyadic productof two vectorsA andB is given by

! = AB = (A1q1 + A2q2 +A3q3)(B1q1 + B2q2 + B3q3) (M1.30)

It is seen that the vectors are not separated by a dot or a cross. At first glance thistype of vector product seems strange. However, the advantage of this notation will


Fig. M1.6 Geometric representation of the scalar triple product.

become apparent later. On performing the dyadic multiplication we obtain

! = AB = A1B1q1q1 + A1B2q1q2 +A1B3q1q3+ A2B1q2q1 +A2B2q2q2 +A2B3q2q3+ A3B1q3q1 +A3B2q3q2 +A3B3q3q3

(M1.31)

In carrying out the general multiplication, we must be careful not to change theposition of the basis vectors. The following statements are valid:

(A + B)C = AC + BC, AB �= BA (M1.32)

M1.3.4 The scalar triple product

The scalar triple product, sometimes also called the box product, is defined by

A · (B × C) = [A,B,C] (M1.33)

The absolute value of the scalar triple product measures the volume of the paral-lelepiped having the three vectorsA, B,C as adjacent edges, see Figure M1.6. Theheighth of the parallelepiped is found by projecting the vectorA onto the crossproductB × C. If the volume vanishes then the three vectors are coplanar. Thissituation will occur whenever a vector appears twice in the scalar triple product. Itis apparent that, in the scalar triple product, any cyclic permutation of the factorsleaves the value of the scalar triple product unchanged. A permutation that reversesthe original cyclic order changes the sign of the product:

[A,B,C] = [B,C,A] = [C,A,B]

[A,B,C] = −[B,A,C] = −[A,C,B](M1.34)


From these observations we may conclude that, in any scalar triple product, thedot and the cross can be interchangedwithout changing themagnitude and the signof the scalar triple product

A · (B × C) = (A × B) ·C (M1.35)

For the general vector basis the coordinate representation of the scalar tripleproduct yields

A· (B×C) = (A1q1+A2q2+A3q3) ·

∣∣∣∣∣∣∣∣∣

q2 × q3 q3 × q1 q1 × q2B1 B2 B3

C1 C2 C3

∣∣∣∣∣∣∣∣∣(M1.36)

It is customary to assign the symbol√g to the scalar triple product of the basis

vectors: √g = q1 · q2 × q3 (M1.37)

It is regrettable that the symbolg is also assigned to the acceleration due to gravity,but confusion is unlikely to occur. By combining equations (M1.36) and (M1.37)we obtain the following important form of the scalar triple product:

[A,B,C] = √g

∣∣∣∣∣∣∣∣∣

A1 A2 A3

B1 B2 B3

C1 C2 C3

∣∣∣∣∣∣∣∣∣(M1.38)

For the basis vectors of the Cartesian system we obtain from (M1.37)

√g = i · (j × k) = 1 (M1.39)

so that in the Cartesian coordinate system (M1.38) reduces to

[A,B,C] =

∣∣∣∣∣∣∣∣∣

Ax Ay Az

Bx By Bz

Cx Cy Cz

∣∣∣∣∣∣∣∣∣(M1.40)

In this expression, according to equation (M1.10), the componentsA1, A2, A3,

etc. have been written asAx,Ay,Az.


Without proof we accept the formula

[A,B,C]2 =

∣∣∣∣∣∣∣∣∣

A · A A · B A ·CB · A B · B B · CC · A C · B C ·C

∣∣∣∣∣∣∣∣∣(M1.41)

which is known as theGram determinant. The proof, however, will be given later.Application of this important formula gives

[q1,q2,q3]2 = (√g)2 =

∣∣∣∣∣∣∣∣∣

q1 · q1 q1 · q2 q1 · q3q2 · q1 q2 · q2 q2 · q3q3 · q1 q3 · q2 q3 · q3

∣∣∣∣∣∣∣∣∣=

∣∣∣∣∣∣∣∣∣

g11 g12 g13

g21 g22 g23

g31 g32 g33

∣∣∣∣∣∣∣∣∣= |gij |

(M1.42)

which involves all elementsgij of the metric tensor. Comparison of (M1.37) and(M1.42) yields the important statement

q1 · (q2 × q3) = √g = √|gij | (M1.43)

so that the scalar triple product involving the general basis vectorsq1,q2,q3 caneasily be evaluated. This will be done in some detail when we consider variouscoordinate systems. Owing to (M1.43),

√g is called thefunctional determinantof

the system.

M1.3.5 The vectorial triple product

At this point it will be sufficient to state the extremely important formula

A × (B× C) = (A · C)B− (A · B)C (M1.44)

which is also known as theGrassmann rule. It should be noted that, without theparentheses, the meaning of (M1.44) is not unique. The proof of this equation willbe given later with the help of the so-called reciprocal coordinate system.


M1.3.6 The scalar product of a vector with a dyadic

On performing the scalar product of a vector with a dyadic we see that the com-mutative law is not valid:

D = A · (BC) = (A · B)C, E = (BC) · A = B(C · A) (M1.45)

Whereas in the first expression the vectorsD andC are collinear, in the secondexpression the direction ofE is along the vectorB so thatD �= E.

M1.3.7 Products involving four vectors

Let us consider the expression (A × B) · (C× D). Defining the vectorF = C×Dwe obtain the scalar triple product

(A × B) · (C× D) = (A × B) · F = A · (B× F) = A · [B× (C× D)] (M1.46)

This equation results from interchanging the dot and the cross and by replacing thevectorF by its definition. Application of the Grassmann rule (M1.44) yields

(A × B) · (C×D) = A · [(B ·D)C− (B · C)D] = (A · C)(B ·D) − (A ·D)(B ·C)(M1.47)

so that equation (M1.46) can be written as

(A × B) · (C× D) =∣∣∣∣∣∣A · C A ·DB · C B · D

∣∣∣∣∣∣ (M1.48)

The vector product of four vectors may be evaluated with the help of the Grass-mann rule:

(A × B) × (C× D) = (F ·D)C− (F · C)D with F = A × B (M1.49)

On replacingF by its definition and using the rules of the scalar triple product, wefind the following useful expression:

(A × B) × (C× D) = [A,B,D]C − [A,B,C]D (M1.50)

M1.4 Reciprocal coordinate systems 15

M1.4 Reciprocal coordinate systems

As will be seen shortly, operations with the so-called reciprocal basis systemsresult in particularly convenient mathematical expressions. Let us consider twobasis systems. One of these is defined by the three linearly independent basisvectorsqi , i = 1,2,3, and the other one by the linearly independent basis vectorsqi , i = 1,2,3. To have reciprocality for the basis vectors the following relationmust be valid:

qi · qk = qk · qi = δki with δki ={0 i �= k1 i = k

(M1.51)

whereδki is the Kronecker-delta symbol. Reciprocal systems are also calledcon-tragredient systems. As is customary, the system represented by basis vectors withthe lower index is calledcovariantwhile the system employing basis vectors withan upper index is calledcontravariant. Therefore,qi andqi are calledcovariantandcontravariant basis vectors, respectively.

Consider for example in (M1.51) the casei = k = 1. While the scalar productq1 · q1 = 1 may be viewed as a normalization condition for the two systems, thescalar productsq1· q2 = 0 andq1· q3 = 0 are conditions of orthogonality. Thus,q1is perpendicular toq2 and toq3 so that we may write

q1 = C(q2 × q3) (M1.52a)

whereC is a factor of proportionality. On substituting this expression into thenormalization condition we obtain forC

q1 · q1 = Cq1 · (q2 × q3) = 1 =⇒ C = 1

q1 · (q2 × q3)(M1.52b)

so that (M1.52a) yields

q1 = q2 × q3

[q1,q2,q3](M1.52c)

We may repeat this exercise withq2 andq3 and find the general expression

qi = qj × qk

[q1,q2,q3](M1.53)

with i, j, k in cyclic order. Similarly we may write forq1, with D as the propor-tionality constant,

q1 = D(q2×q3), q1·q1 = Dq1· (q2×q3) = 1 =⇒ q1 = q2 × q3[q1,q2,q3]

(M1.54)


Thus, the general expression is

qi = qj × qk[q1,q2,q3]

(M1.55)

with i, j, k in cyclic order. Equations (M1.53) and (M1.55) give the explicit ex-pressions relating the basis vectors of the two reciprocal systems.

Let us consider the special case of the Cartesian coordinate system with basisvectorsi1, i2, i3. Application of (M1.55) shows thatij = ij since [i1, i2, i3] = 1, sothat in the Cartesian coordinate system there is no difference between covariant andcontravariant basis vectors. This is the reason why we have writtenAi = Ai, i =1,2,3 in (M1.10).

Now we return to equation (M1.43). By replacing the covariant basis vectorq1with the help of (M1.52c) and utilizing (M1.48) we find

q1 · (q2 × q3) = (q2 × q3) · (q2 × q3)[q1,q2,q3]

= 1

[q1,q2,q3]

∣∣∣∣∣∣q2 · q2 q2 · q3q3 · q2 q3 · q3

∣∣∣∣∣∣ = 1

[q1,q2,q3]

(M1.56)

From (M1.51) it follows that the value of the determinant in (M1.56) is equal to 1.Sinceq1 · (q2 × q3) = √

g we immediately find

[q1,q2,q3] = 1√g

(M1.57)

Thus, the introduction of the contravariant basis vectors shows that (M1.43) and(M1.57) are inverse relations.

Often it is desirable to work with unit vectors having the same directions as theselected three linearly independent basis vectors. The desired relationships are

ei = qi|qi| = qi√

qi · qi = qi√gii, ei = qi

|qi| = qi√qi · qi = qi√

gii(M1.58)

While the scalar product of the covariant basis vectorsqi · qj = gij defines theelements of thecovariant metric tensor, thecontravariant metric tensoris definedby the elementsqi · qj = gij , and we have

qi · qj = qj · qi = gij = gji (M1.59)

M1.4 Reciprocal coordinate systems 17

Owing to the symmetry relationsgij = gji andgij = gji each metric tensor iscompletely specified by six elements.

Some special cases follow directly from the definition (M1.13) of the scalarproduct. In case of an orthonormal system, such as the Cartesian coordinate system,we have

gij = gji = gij = gji = δj

i (M1.60)

As will be shown later, for any orthogonal system the following equation applies:

giigii = 1 (M1.61)

While in the Cartesian coordinate system the metric fundamental quantities areeither 0 or 1, we cannot give any information about thegij or gij unless thecoordinate system is specified. This will be done later when we consider variousphysical situations.

In the following we will give examples of the efficient use of reciprocal systems.Work is defined by the scalar productdA = K · dr , whereK is the force anddr isthe path increment. In the Cartesian system we obtain a particularly simple result:

K ·dr = (Kx i + Ky j + Kzk) · (dx i + dy j + dz k) = Kx dx + Ky dy + Kz dz

(M1.62)

consisting of three work contributions in the directions of the three coordinate axes.For specific applications it may be necessary, however, to employ more generalcoordinate systems. Let us consider, for example, an oblique coordinate systemwith contravariant components and covariant basis vectors ofK anddr . In thiscase work will be expressed by

K · dr = (K1q1 +K2q2 +K3q3) · (dq1q1 + dq2q2 + dq3q3)

= Km dqn qm · qn = Km dqn gmn(M1.63)

Expansion of this expression results in nine components in contrast to only threecomponents of the Cartesian coordinate system. A great deal of simplification isachieved by employing reciprocal systems for the force and the path increment.As in the case of the Cartesian system, work can then be expressed by using onlythree terms:

K · dr = (K1q1 +K2q2 +K3q3) · (dq1 q1 + dq2 q2 + dq3 q3)

= Km dqn qm · qn = Km dq

n δmn = K1 dq1 +K2 dq

2 +K3 dq3

(M1.64a)


or

K · dr = (K1q1 +K2q2 +K3q3) · (dq1 q1 + dq2q2 + dq3 q3)

= Km dqn qm · qn = Km dqn δnm = K1 dq1 +K2 dq2 +K3 dq3

(M1.64b)

Finally, utilizing reciprocal coordinate systems, it is easy to give the proof ofthe Grassmann rule (M1.44). Let us consider the expressionD = A × (B × C).According to the definition (M1.26) of the vector product,D is perpendicular toAand to (B× C). Therefore,D must lie in the plane defined by the vectorsB andCso that we may write

A × (B×C) = λB+ µC (M1.65)

whereλ andµ are unknown scalars to be determined. Tomake use of the propertiesof the reciprocal system,we first setB = q1 andC = q2. These two vectors define aplane oblique coordinate system. To complete the systemwe assume that the vectorq3 is a unit vector orthogonal to the plane spanned byq1 andq2. Thus, we have

B = q1, C = q2, e3 = q1 × q2|q1 × q2| (M1.66)

and

q1 · (q2 × e3) = e3 · (q1 × q2) = e3 · e3 |q1 × q2| = |q1 × q2| (M1.67)

According to (M1.55), the coordinate system which is reciprocal to the (q1,q2,q3)system is given by

q1 = q2 × e3|q1 × q2| , q2 = e3 × q1

|q1 × q2| , e3 = q1 × q2|q1 × q2| = e3 (M1.68)

The determination ofλ andµ follows from scalar multiplication ofA× (B×C) =A × (q1 × q2) = λq1 + µq2 by the reciprocal basis vectorsq1 andq2:

λ = [A × (q1 × q2)

] · q1 = A × (q1 × q2) · (q2 × e3)|q1 × q2|

= (A × e3) · (q2 × e3) = A · q2 = A · C(M1.69a)

Analogously we obtain

µ = [A × (q1 × q2)

] · q2 = (A × e3) · (e3 × q1) = −A · q1 = −A ·B (M1.69b)

Substitution ofλ andµ into (M1.65) gives the final result

A × (B× C) = (A · C)B− (A · B)C (M1.70)

M1.5 Vector representations 19

M1.5 Vector representations

The vectorA may be represented with the help of the covariant basis vectorsqi orei and the contravariant basis vectorsqi or ei as

A = Amqm = Amqm = A* mem = A

*

mem (M1.71)

The invariant character ofA is recognizedby virtue of the fact thatwe have the samenumber of upper and lower indices. In addition to thecontravariantandcovariantmeasure numbersAi andAi of the basis vectorsqi andqi we have also introducedthephysical measure numbersA

*i andA

*

i of the unit vectorsei andei. In generalthe contravariant and covariant measure numbers do not have uniform dimensions.This becomesobviouson considering, for example, the spherical coordinate systemwhich is definedby two angles, which aremeasured in degrees, and the radius of thesphere, which is measured in units of length. Physical measure numbers, however,are uniformly dimensioned. They represent the lengths of the components of avector in the directions of the basis vectors. The formal definitions of the physicalmeasure numbers are

A* i = Ai |qi| = Ai√gii, A

*

i = Ai |qi| = Ai

√gii (M1.72)

Nowwewill showwhat consequencesarise by interpreting themeasure numbersvectorially. Scalar multiplication ofA = Anqn by the reciprocal basis vectorqi

yields forAi

A · qi = Amqm · qi = Amδim = Ai (M1.73)

so thatA = Amqm = A · qmqm (M1.74)

This expression leads to the introduction to theunit dyadicE,

E = qmqm (M1.75a)

This very special dyadic or unit tensor of rank two has the same degree of im-portance in tensor analysis as the unit vector in vector analysis. The unit dyadicE is indispensable and will accompany our work from now on. In the Cartesiancoordinate system the unit dyadic is given by

E = ii + jj + kk = i1i1 + i2i2 + i3i3 (M1.75b)

We repeat the above procedure by representing the vectorA asA = Amqm.Scalar multiplication byqi results in

A · qi = Amqm · qi = Ai (M1.76)


and the equivalent definition of the unit dyadicE

A = Amqm = A · qmqm =⇒ E = qmqm (M1.77)

Of particular interest is the scalar product of two unit dyadics:

E · E = qmqm · qnqn = qmδnmqn = qmqm = E

E · E = qmqm · qnqn = qmδmn qn = qmqm = E

(M1.78)

From these expressions we obtain additional representations of the unit dyadic thatinvolve the metric fundamental quantitiesgij andgij :

E · E = qmqm · qnqn = gmnqmqn = qmqm · qnqn = gmnqmqn (M1.79)

Again it should be carefully observed that eachexpression containsanequal numberof subscripts and superscripts to stress the invariant character of the unit dyadic.We collect the important results involving the unit dyadic as

E = qmqm = δnmqmqn = qmqm = δmn qmq

n = gmnqmqn = gmnqmqn (M1.80)

Scalar multiplication ofE in two of the forms of (M1.80) withqi results in

E · qi = (qmqm) · qi = qmδmi = qi= (gmnqmqn) · qi = gmnqmδni = gimqm

(M1.81)

Hence, we see immediately that

qi = gimqm (M1.82)

This very useful expression is known as theraising rule for the index of the basisvectorqi. Analogously we multiply the unit dyadic byqi to obtain

E · qi = (qmqm) · qi = qi = (gmnqmqn) · qi = gimqm (M1.83)

and thusqi = gimqm (M1.84)

which is known as thelowering rule for the index of the contravariant basisvectorqi.

With the help of the unit dyadic we are in a position to find additional importantrules of tensor analysis. In order to avoid confusion, it is often necessary to replace aletter representing a summation index by another letter so that the letter representinga summation does not occur more often than twice. If the replacement is done

M1.5 Vector representations 21

properly, the meaning of any mathematical expression will not change. Let usconsider the expression

E = qrqr = grmgrnqmqn = δnmq

mqn (M1.85)

Application of (M1.82) and (M1.84) gives the expression to the right of the secondequality sign. For comparison purposes we have also added one of the forms of(M1.80) as the final expression in (M1.85). It should be carefully observed that thesummation indicesm,n, r occur twice only.

To take full advantage of the reciprocal systems we perform a scalar multiplica-tion first by the contravariant basis vectorqi and then by the covariant basis vectorqj , yielding

(E · qi) · qj = grmgrnδinδ

mj = δnmδ

inδmj (M1.86)

from which it follows immediately that

grjgri = δij (M1.87a)

By interchangingi andj , observing the symmetry of the fundamental quantities,we find

girgrj = δ

j

i or (gij )(gij ) =

1 0 0

0 1 0

0 0 1

=⇒ (gij ) = (gij )−1 (M1.87b)

Hence, thematrices (gij ) and (gij ) are inverse to each other. Owing to the symmetryproperties of the metric fundamental quantities, i.e.gij = gji andgij = gji , weneed six elements only to specify either metric tensor. In case of an orthogonalsystemgij = 0, gij = 0 for i �= j so that (M1.87a) reduces to

giigii = 1 (M1.88)

thus verifying equation (M1.61). At this point we must recall the rule that we donot sum over repeated free indicesi, j, k, l.

Next we wish to show that, in an orthonormal system, there is no differencebetween contravariant and covariant basis vectors. The proof is very simple:

ei = qi√gii

= gin√giiqn = √

giiqi = qi√gii

= ei (M1.89)

Here use of the raising rule has been made. With the help of (M1.89) it is easyto show that there is no difference between contravariant and covariant physicalmeasure numbers. Utilizing (M1.71) we find

A · ei = A · ei =⇒ A* nen · ei = A

*

nen · ei =⇒ A* i = A

*

i (M1.90)


M1.6 Products of vectors in general coordinate systems

There are various ways to express the dyadic product of vectorA with vectorB byemploying covariant and contravariant basis vectors:

AB = AmBnqmqn = AmBnqmqn = AmBnqmqn = AmBnqmqn (M1.91)

This yields four possibilities for formulating the scalar productA · B:A · B = AmBnqm · qn = AmBngmn = AmBnqm · qn = AmBng

mn

= AmBnqm · qn = AmB

m = AmBnqm · qn = AmBm(M1.92)

1 The last two forms with mixed basis vectors (covariant and contravariant) aremore convenient since the sums involve the evaluation of only three terms. Incontrast, nine terms are required for the first two forms since they involve themetric fundamental quantities.

There are two useful forms in which to express the vector productA × B. Fromthe basic definition (M1.28) and the properties of the reciprocal systems (M1.55)we obtain

A × B = Amqm × Bnqn = √g

∣∣∣∣∣∣∣∣∣

q1 q2 q3

A1 A2 A3

B1 B2 B3

∣∣∣∣∣∣∣∣∣(M1.93)

where all measure numbers are of the contravariant type. If it is desirable toexpress the vector product in terms of covariant measure numbers we use (M1.53)and (M1.57). Thus, we find

A × B = Amqm × Bnqn = 1√g

∣∣∣∣∣∣∣∣∣

q1 q2 q3A1 A2 A3

B1 B2 B3

∣∣∣∣∣∣∣∣∣(M1.94)

The two forms involving mixed basis vectors are not used, in general.On performing the scalar triple product operation (M1.33) we find

[A,B,C] = Amqm · (Bnqn × Crqr)

= Am√gqm ·

∣∣∣∣∣∣∣∣∣

q1 q2 q3

B1 B2 B3

C1 C2 C3

∣∣∣∣∣∣∣∣∣= √

g

∣∣∣∣∣∣∣∣∣

A1 A2 A3

B1 B2 B3

C1 C2 C3

∣∣∣∣∣∣∣∣∣(M1.95)

1 For the scalar productA ·A we usually writeA2.

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