Coastal Lagoons - Chapter 3 pdf

3.2.2 Advective Flux3.2.3 Diffusive Flux3.2.4 Elementary Area and Elementary Volume3.2.5 Net Flux across a Closed Surface3.3 Transport and Evolution3.3.1 Rate of Accumulation3.3.2 Lagran

3.2.2 Advective Flux3.2.3 Diffusive Flux3.2.4 Elementary Area and Elementary Volume3.2.5 Net Flux across a Closed Surface3.3 Transport and Evolution

3.3.1 Rate of Accumulation3.3.2 Lagrangian Form of the Evolution Equation3.3.3 Eulerian Form of the Evolution Equation3.3.4 Differential Form of the Transport Equation3.3.5 Boundary and Initial Conditions

3.4 Hydrodynamics3.4.1 Conservation Laws in Hydrodynamics3.4.1.1 Conservation of Mass3.4.1.2 Conservation of Momentum

3.4.1.2.1 The Euler Equations3.4.1.2.2 The Euler Equations in a Rotating

Frame or Reference3.4.1.2.3 The Navier-Stokes Equations3.4.1.3 Conservation of Energy

3.4.1.4 Conservation of Salt3.4.1.5 Equation of State3.4.2 Simplification and Scale Analysis3.4.2.1 Incompressibility3.4.2.2 The Hydrostatic Approximation3.4.2.3 The Coriolis Force

3.4.2.4 The Reynolds Equations3.4.2.5 The Primitive Equations3.4.3 Special Flows and Simplifications in Dimensionality3.4.3.1 Barotrophic 2D Equations

3.4.3.2 1D Equations (Channel Flow)3

3.4.4 Initial and Boundary Conditions3.4.4.1 Initial Conditions3.4.4.2 Conditions on Material Boundaries3.4.4.3 Conditions on Open Boundaries3.4.4.4 Conditions on the Sea Surface and the Sea Bottom3.5 Boundary Processes

3.5.1 Bottom Processes3.5.1.1 Bottom Shear Stress3.5.1.2 Other Bottom Processes3.5.2 Solid Boundary Processes3.5.3 Free Surface Processes3.5.3.1 Mass Exchange3.5.3.2 Momentum Exchange3.5.3.3 Energy Exchange3.5.4 Cohesive and Noncohesive Sediment ProcessesBibliography

3.1 INTRODUCTION TO TRANSPORT PHENOMENA

Chapter 2 concluded that the calculation of spatio-temporal distribution of majorcomponents of a lagoon’s hydrogeomorphological unit and biocoenose is importantfor the description of the structure and function dynamics (productivity and carryingcapacity) of the lagoon system and, consequently, for sustainable management Thisconcept is required to understand the transport phenomena that describe the evolution

of properties due to fluid motion (advection) and/or molecular and turbulent ics (diffusion) In the case of turbulent flows the small-scale motion of the fluidparticles is actually random, and this nonresolved advection is also treated as diffu-sion (eddy diffusion)

dynam-A mathematical description of the transport phenomena (transport equations) isbased on the concept of conservation principle, which is valid in any application.Conservation principle can be stated as

{The rate of accumulation of a property inside a control volume}

= {what flows in minus what flows out}+{production minus consumption}Using this conservation principle, transport equations for any property inside acontrol volume can be derived if production and consumption mechanisms are knownand if the control volume and transport processes are quantified The control volume

is presented as the largest volume for which one can consider the interior properties

as uniformly distributed as well as fluxes across the surface

In previous coastal lagoon studies the control volume was often implicitlydefined as the whole lagoon Concepts of residence and flushing time were derivedfrom this global approach (see Chapter 5 for details) In that case, only fluxes atthe boundaries were required This type of integral approach cannot describegradients and is consequently not sufficient to support process-oriented research

or management For these purposes the system has to be divided into homogeneousparts (control volumes) and fluxes between them have to be calculated Thisapproach requires a numerical model The number of space dimensions required

to describe the control volumes equals the number of dimensions of the model.The resolution of the transport equations in practical situations has been madepossible using numerical methods and computers (see Chapter 6 for details)

Before the advent of computers transport processes had to be studied using ical formulations (derived from experiments) or analytical solutions in simplegeometries or boundary conditions

empir-This chapter presents a general transport equation (also called an evolution tion) based on the concepts of (1) control volume, (2) advective flux, and (3) diffusiveflux Based on this generic equation, equations for hydrodynamics, temperature, salin-ity, and suspended sediments are also introduced Special flows and simplification ofdimensionality and boundary processes and conditions, particularly for coastallagoons, are described in detail for use in lagoon modeling studies

equa-3.2 FLUXES AND TRANSPORT EQUATION 3.2.1 V ELOCITY AND D IFFUSIVITY IN L AMINAR AND T URBULENT

F LOWS AND IN A N UMERICAL M ODEL

For transport purposes, fluids are considered a continuum system Velocity is defined

in a macroscopic way based on the concept of continuum system Because fluidsare not a real continuum, system velocity cannot describe transport processes at themolecular scale The nonrepresented processes are represented by diffusion.Although the concept of velocity is well known, it is reconsidered for modelingpurposes

Diffusion in laminar flows occurs from movements at a molecular scale notrepresented by the velocity In turbulent flows, velocity, as defined for laminar flows,becomes time dependent, changing at a frequency that is too high to be representedanalytically As a consequence time average values must be considered, followingthe Reynolds approach (see Section 3.4.2.4 for details) Transport processes, notdescribed by this average velocity, are represented by turbulent diffusion (using aneddy diffusivity, which is several orders of magnitude higher than molecular diffu-sivity) More information on this topic is given in Section 3.4

Most numerical models use grids with spatial and time steps larger than thoseassociated with turbulent eddies Again, processes not resolved by velocity computed

by models have to be accounted for by diffusion (subgrid diffusion)

The box represented in Figure 3.1 is commonly used to illustrate the concept ofdiffusion in laminar flows The same box could be used to illustrate the concept ofeddy diffusion in turbulent flows or subgrid diffusion in numerical models Inmolecular diffusion white and black dots represent molecules, while in other casesthey represent eddies In the initial conditions (stage (a) in Figure 3.1) two differentfluids are kept apart by a diaphragm Molecules inside each half-box move randomly,with velocities not described by our model (Brownian or eddy) When the diaphragm

is removed, particles from each side keep moving, resulting in the possibility of

mixing (stage (b) in Figure 3.1) After some time the proportion of white and darkfluid is the same in both the box halves At this stage, the probability of a whitemolecule moving from the right side of the box to the left side is equal to theprobability of another white molecule moving the opposite way and, therefore, there

is no net exchange (stage (c) in Figure 3.1)

A macroscopic view of the mixing is represented in Figure 3.2 In the initialcondition (stage (a) in Figure 3.2), a black and a white side is observed (stage (b)

in Figure 3.2); during the mixing process a growing gray area occurs, corresponding

to the mixing zone; and after complete mixing, a homogeneous gray fluid is observed(stage (c) in Figure 3.2)

The velocity of each elementary portion of fluid, like the velocity of any othermaterial point, is defined using two consecutive locations, as shown in Figure 3.3:

(3.1)

Knowing the velocity of each individual molecule, it will be possible to fullycharacterize transport But, according to Heisenberg’s uncertainty principle, it willnever be possible to know the place and the velocity of each molecule simultaneously.Consequently, the fluid has to be considered a continuum system, for which a velocity

is defined

FIGURE 3.1 Distribution of molecules of two different substances, inside a box at three different moments: initially kept apart by a diaphragm (a), during mixing (b), and when spatial gradient has disappeared (c).

FIGURE 3.2 Macroscopic view of the fluid composed by the molecules represented in Figure 3.1.

Considering a fluid as a continuum system, an elementary volume of fluid is aportion of fluid so small that its properties (including velocity) can be considered

as uniform, but it is much bigger than the size of a molecule in laminar flow andthan an eddy in turbulent flow In the case of a numerical model, an elementaryvolume is the volume included inside a grid cell As a consequence, diffusivityincreases with grid size in numerical models

3.2.2 A DVECTIVE F LUX

The advective flux accounts for the amount of a property transported per unit oftime due to fluid velocity across a surface perpendicular to the motion Its dimensionsare [BT−1] and can be expressed as†

(3.2)

where V is the volume The quantity β=B/V has the dimensions of a specific quantity(amount per unit of volume) and is called the concentration The ratio between thevolume and the time [L3T−1] is the flow rate that can be calculated as the product

FIGURE 3.3 Trajectory of an elementary portion of fluid showing consecutive locations apart

=

of the velocity [LT−1] and the area [L2] of the surface The transport produced bythe velocity per unit of area is

where is the velocity relative to the velocity of the surface This flux is a vectorparallel to the velocity The flux across an elementary area dA not perpendicular tothe velocity is given by

where is the external normal to the elementary surface dA In the case of a finitesurface A, the total flux is the summation of the elementary fluxes across theelementary areas composing it This summation can be represented by the integralover that surface:

(3.4)

where ϕ is the diffusivity and the quantity inside the parentheses is the gradient of

β, the specific value of B (concentration in case of mass)

Both β gradient and diffusivity can vary spatially, implying the calculation ofthe flux on elementary surfaces:

and its integration along the overall surface:

n

3.2.4 E LEMENTARY A REA AND E LEMENTARY V OLUME

The transport across a finite surface A is obtained as the integration of the transport

across elementary areas where properties can be assumed to be uniform

Mathemati-cally, elementary areas are infinitesimal, but physically they are just small enough

to allow the assumption that properties assume constant values on their surfaces,

allowing for the substitution of the integrals by summations Fluxes across an

elementary surface are obtained by multiplying fluxes per unit of area with the area

of the elementary surface This is a basic assumption of modeling

An elementary volume is a volume limited by elementary areas, inside whichproperties can be considered as having uniform values The total amount of a

property contained inside an elementary volume is given by the product of its specific

value with its elementary volume

3.2.5 N ET F LUX ACROSS A C LOSED S URFACE

Let us consider a closed surface as represented in Figure 3.4 In the figure an

elementary area ∆A, local normal n, and velocity u, which can vary from point to

point, are represented In regions where normal and velocity have opposite senses

the internal product is negative, meaning that property B is being advected

(trans-ported) into the interior of the volume limited by the surface Where the internal

product is positive, the property is being transported outward Thus, the integral of

the flux (advective or diffusive) over a closed surface gives the difference between

the amount of property being transported outward and inward

3.3 TRANSPORT AND EVOLUTION

An evolution equation describes the transformations suffered by a property as time

progresses The properties of a fluid limited by solid surface can be modified only

by production (sources) or destruction (sinks) processes If the boundary of the

FIGURE 3.4 Representation of a generic transparent surface, an elementary area on that

surface, the respective exterior normal, and the fluid velocity.

volume is permeable, allowing for advective and/or diffusive fluxes, transport also

will contribute for the evolution of the property inside the volume

3.3.1 R ATE OF A CCUMULATION

The rate of accumulation of a property B inside an elementary volume is the ratio

between the variation of the total amount contained in the volume and the time

interval during which accumulation happened This concept can be translated by the

algebraic equation:

In this equation, the total amount of B inside an elementary volume V in each moment

is given by the product of the specific value β multiplied with the volume.† If an

infinitesimal time interval is considered, the previous equation can be written in a

differential form:

This equation puts into evidence the physical meaning of the time derivative, showing

that it describes the rate of accumulation

3.3.2 LAGRANGIAN FORM OF THE EVOLUTION EQUATION

If there is no flux across the boundary of the control volume, the rate of accumulation

accounts for the sources (S 0 ) minus the sinks (S i):

This can happen in the case of a volume limited by a solid boundary or in the case of

a volume moving at the same velocity as the flow where diffusivity can be neglected

Faecal bacteria are traditionally assumed to be unable to grow in saline waterand have a first-order decay rate For such a variable in a volume where advective

and diffusive fluxes are null and the sink is the mortality of bacteria, the evolution

equation would be written as

where m is the rate of mortality.

† In a general case, the total amount inside a volume V of a property with a specific value β is the integral

of β inside the volume In the case that β is constant, the integral is the product of β times the volume.

If the surface limiting the volume is permeable, a diffusive flux exists tional to the gradient and the evolution equation becomes

propor-(3.6)

Please note that a positive flux is directed out of the elementary volume and is,therefore, lowering the mass of the property inside the control volume The negativesign in front of the integral accounts for this fact

Equation (3.6) describes the evolution of a generic property B, with a specific

value β, if there is no advective exchange between the elementary volume of fluidand the surrounding volume That is the case when the elementary volume moves

at the same velocity as the fluid (null relative velocity), corresponding to theLagrangian formulation of the problem Measuring instruments moving freely, trans-ported by the flow (e.g., attached to a buoy), would carry out this type of measure-ment That is not, however, the common way of measuring

3.3.3 EULERIAN FORM OF THE EVOLUTION EQUATION

In general, field measurements are carried out in fixed stations (see Chapter 7 fordetails) If such a monitoring strategy is adopted, the elementary volume to beconsidered in the evolution equation is fixed in space, and the velocity relative tothe surface of the volume becomes the flow velocity In that case the evolutionequation becomes:

(3.7)

The partial derivative states that the control volume does not move and, consequently,the velocity to be considered in the advective flux is the flow velocity This equationholds for situations where the volume is time dependent If we consider the case of

a rigid volume, we can take it out of the time derivative

3.3.4 DIFFERENTIAL FORM OF THE TRANSPORT EQUATION

Let us consider a Cartesian reference and an elementary cubic volume, as represented

in Figure 3.5 Being an elementary volume, it is small enough to assume thatproperties have uniform values on the surface and that the value of the property can

be considered uniform inside the volume Considering this approach and the metric properties indicated in the figure, we can write an equation for the volumeand for the fluxes The volume is given by

corresponding surface Keeping the orientation of the normal in mind, we can

write the fluxes for surfaces perpendicular to the x1 axis:

Doing a similar calculation for other directions, dividing the whole equation by ∆V

and letting ∆x i converge to zero, we obtain Equation (3.8):

(3.8)

In this equation the Einstein convention† for summation is used and sources andsinks are calculated per unit of volume Equation (3.7) and Equation (3.8) are generaland hold for any property They can also be used to derive continuity and momentumequations in the next sections

FIGURE 3.5 Cubic type elementary control volume Velocity components are represented

on every surface of the volume.

† In the Einstein convention, a doubled index represents a summation of the terms obtained replacing that index by each dimension of the physical space (3 in a three-dimensional space).

1 1

Following a similar procedure and assuming an incompressible fluid—an tion equivalent to Equation (3.8) on a Lagrangian reference can be obtained fromEquation (3.6):

equa-(3.9)

Comparing Equation (3.8) and Equation (3.9) and using the fact that for an pressible fluid ∂u i/∂x i= 0 (see Section 3.4.1.1), the relation between total and partialtime derivatives is obtained:†

incom-(3.10)

This equation simply says that the property of a fixed elementary volume in a movingfluid element can change through local changes in the fluid and advective changestransported by the fluid into the elementary volume

In the field of mathematical modeling, especially physical processes, there is astrong tradition of obtaining the discretized equations starting from their differentialform using, for example, the Taylor series In this text discretized equations will beobtained using both a finite-volume approach and the Taylor series

3.3.5 BOUNDARY AND INITIAL CONDITIONS

Partial differential equations relate values of properties to time and space derivatives.Spatial derivatives relate values of the property in neighboring points, and conse-quently the evaluation at boundaries requires the knowledge of information fromoutside the study area called boundary conditions Similarly, time derivative relatesvalues of the property in sequential instants of time and their evaluation requiresknowledge of the solution in the first instant of time called initial conditions.Initial conditions have to be specified in terms of property values In contrast,spatial boundary conditions can be specified in terms of values or their derivatives.Comparing differential and integral forms of evolution equations (Equation (3.7)and Equation (3.8)) shows that imposing boundary conditions in terms of spatialderivatives in differential equations is, in fact, equivalent to imposing fluxes acrossthe boundary in integral equations

Physically boundaries can be divided into two main groups: solid boundariesand open boundaries Solid boundaries are impermeable and, consequently, there

is neither water flux nor advective transport across them Diffusive flux acrosssolid boundaries can be neglected for most dissolved substances, but not formomentum, where it is represented by bottom shear stress Solid matter can be

† In fact, the condition of null velocity divergence is not required, only the continuity equation What happens is that the continuity equation becomes the condition of zero velocity divergence in the case of incompressible flows.

deposited or eroded from the bottom, resulting in apparent fluxes across that solidboundary.

In tidal flows some regions are covered and some are uncovered depending

on the level of water The boundary between covered and uncovered regions is

a moving, solid boundary, and its location has to be calculated by the model ateach time step

Open boundaries are artificial boundaries and usually require elaborate lations At open boundaries, conditions have to be specified using measurements orestimated using the solution calculated inside the domain (radiative boundaries)

formu-The most complex situation at open boundaries is the case of active properties

properties that modify the flow This is the case with baroclinic flows, wheregradients of water density generate density flows In this case, any boundarycondition error generates a force that modifies the flow itself, originating a quickpropagation of the error to inner points This can be extremely important forcoastal lagoons

As a general rule, the diffusive flux is much smaller than the advective flux andcan be neglected at the open boundary Doing so, the boundary values of propertieshave to be specified only when the flow enters into the computation area, as in thecase of coastal lagoons The net advective transport is proportional to local gradientsand to local transport As a consequence, in the case of passive tracers, openboundaries must be located in regions with small velocities and small gradients.Major problems are found with properties acting directly on flow That is thecase of the sea surface level, which is transported at the speed of gravity waves(proportional to the square root of the water depth), and its local value is the result

of waves propagating in many directions At the boundary, the sea level depends onincoming and outgoing waves

Incoming waves have to be specified and outgoing waves have to be estimatedusing the solution inside the domain and an approach for their propagation speed.This is the basis of a radiation condition The distinction between contributions

of incoming and outgoing waves is a major difficulty, especially in tidal systemswhere measured tide level itself is the result of waves propagating in differentdirections

Free surface can be treated as a solid boundary in the sense that there are noadvective fluxes Across this boundary one can exchange momentum, heat, and mass.Mass of any substance can enter the system through this boundary (e.g., precipita-tion), but only gaseous matter can leave it (e.g., evaporation) Fluxes across thisboundary depend mainly on atmospheric conditions and are proportional to itssurface, being much more important in the ocean than in coastal lagoons

In coastal lagoons the relevance of momentum exchange across free surfacedecreases as tidal amplitude increases In contrast, the exchange of heat across thisboundary tends to be more important than the exchange across the open boundarybecause in coastal lagoons the ratio between surface and volume is much smallerthan that in the sea

In the next sections additional information on boundary conditions will be suppliedfor each specific property It will be seen that the most adequate conditions depend

on the availability of the measuring devices used for obtaining the information

3.4 HYDRODYNAMICS

The hydrodynamic behavior is governed by mathematical equations that have beenknown for more than 100 years It is believed that these equations have universalcharacter, and the only reason that we are not able to exactly predict the dynamicalevolution of a water body is that we do not possess analytical solutions to theseequations and our knowledge of boundary and initial conditions is incomplete Thesepoints will be discussed later in the sections that follow

The basic hydrodynamic equation can be derived in its full form through theapplication of conservation laws to the basic variables of the system More specif-ically, the general conservation equation derived in the last chapter can be directlyapplied to the variables under consideration

In the oceanic environment, including coastal lagoons, there are seven variablesthat completely define the state of the fluid: the water density ρ; the three velocity

components u i ,i = 1,3, in the direction of x i ,i = 1,3; the pressure p; the temperature

T; and the salinity S If only freshwater systems are considered, salinity is not a

variable, reducing the number of state variables to six

Remarkably, the same set of variables is used also for the description of theatmosphere However, in this case, humidity (water vapor content) replaces salinity

as a state variable Moreover, the equations applicable to atmospheric motion differonly slightly from the ones used in the oceans

3.4.1 C ONSERVATION L AWS IN H YDRODYNAMICS

As explained previously, the basic hydrodynamic equations can be relatively easilydeduced from conservation equations of the single state variables A rigorous deduc-tion of these equations will not always be shown, but the equations in their originalform will be presented and their meanings and implications noted It will also beshown how in all possible cases the equations can be derived using the generalconservation equation (Equation (3.8)), noted earlier in this chapter

(3.11)

The conservation equation of mass is also called the continuity equation It expressesthe fact that a mass flux into a fixed volume will lead to an increase of mass in thevolume due to an increase in density

i i

0

If we substitute the partial derivative with the total one, we can rewrite thecontinuity equation as follows:

In this form it is easy to see that for an incompressible fluid (for which dρ/dt = 0

is valid) the continuity equation reduces to:

in neglecting the compressibility effect is that acoustic waves cannot be described

in the water As acoustic waves are of minor importance in oceanographic tions, neglecting these terms is justified

Conservation of momentum is in its simplest form described by Newton’s law

F i= a i m

where F i is the force acting on a fluid volume, a i is the acceleration, and m is the

mass of the fluid particle

Using the density as usual in fluid dynamics instead of the mass of a particle

we can write

where f i is the force per unit volume acting on the fluid volume This equation can

be deduced from the general conservation equation (Equation (3.8)) when applied

to ρu i together with the continuity equation In this case the source and sink terms

are given by f i, and the diffusion terms are neglected for now

3.4.1.2.1 The Euler Equations

The forces acting on a fluid body may be divided conveniently into two classes:the volume forces and the interface forces An example of the first one is gravi-tational force, and an example of the second one is pressure gradient force or windstress

1

0ρ

ρ

d dt

u x

i i

i i

0

ρdu

i i

=

The influence of the pressure gradient forces can be derived in this manner:given an infinitesimal volume ∆V =∆x1∆x2∆x3, the force exerted by the pressure on

the left side of the cube in x1 direction is given by p(x) ∆x2∆x3 (pressure is forceper area) Similarly, the force exerted on the right side of the cube is

Here a Taylor series expansion has been applied to p(x +∆x1) So the net force

on the volume in the x1 direction, after adding the two contributions, is dV,

and the force per unit volume is just The same analysis can be carried outfor the other two directions, giving the total pressure gradient force per unit volume

In this form the equations are called Euler equations, or more precisely, Euler

equations in a nonrotating frame of reference Note that the gravitational acceleration

is a vector with only the vertical component different from zero

g i= (0, 0, −g)

(pointing downward) and g is 9.81 m s−2

3.4.1.2.2 The Euler Equations in a Rotating Frame of Reference

The above-derived equations are not suitable for their application to meso-scale orbasin-wide scale This is because the Earth is not an inertial frame of reference but

is rotating Although this has no impact on water bodies that are small in size, forthe larger applications it is very important to take account of this effect

The influence of the Earth’s rotation can be described with the introduction

of a new apparent volume force called the Coriolis force In this case, the new

forces on the water parcel are composed not only of the pressure gradient, but

x

x g

i i i

= − ∂

∂ +ρ

du dt

p

x g

i

i i

= − ∂

∂ +1ρ

also of the additional Coriolis force, and the new equations (not deduced) can bewritten as

coordinate direction In this form the equations are called Euler equations in a

rotating frame of reference.

Besides the gravitational force, the Coriolis force is the only other importantvolume force that acts in a fluid body Because of the vector product, the Coriolisforce always acts perpendicular to the current velocity, in the northern hemisphere tothe right and in the southern hemisphere to the left of the fluid flow It is the Coriolisforce that is responsible for all the meso-scale structure we can see on the weathercharts that contain cyclones and anticyclones However, the Coriolis force isimportant only for large-scale circulations and, therefore, may not be importantfor many coastal lagoons of the world

3.4.1.2.3 The Navier-Stokes Equations

The above-derived Euler equations describe the flow of a fluid without friction Oftenthis may be a good approximation, especially if no material boundaries (lateral andvertical) are close to the area of investigation Internal friction in a fluid is normallyvery small and the Euler equations provide a satisfactory simplification However,once the fluid is close to a boundary, friction becomes more important and anotherarea force, the stress tensor, has to be introduced A moving fluid layer exerts a force

on the neighboring fluid layers The strength of this force is directly proportional tothe area of the fluid layer and the velocity difference between these layers, andinversely proportional to the distance of the layers:

where A is the area, ∆u1 is the velocity difference from one layer to the other, and

∆x3 is the distance of the fluid layers The parameter µ is the constant of

propor-tionality and is called the dynamic viscosity coefficient It is a parameter that depends

only on the type of fluid and its temperature and salinity contents, but not on thefluid dynamics

du

dt fu

p x

1 2

1

1

− = − ∂

∂ρ

du

dt fu

p x

2 1

2

1+ = − ∂

∂ρ

du dt

F=µ ∆ ∆A u1/ x3

Taking the finite differences to the infinitesimal limit, the stress (force per area)can be written as

Here the subscript 31 means the force exerted by the moving fluid layer in the x1direction due to the velocity gradient in direction x3 Therefore, the stress tensor hasnine components, three terms for every spatial direction

The effect of this stress tensor on the equation of motion can easily be derived.The derivation is almost equivalent to the incorporation of the pressure gradient forceinto the Euler equations, and the result for the viscous force per unit mass in the

x1 direction due to the shear in the x3 direction is

where we have used the kinematic viscosity coefficient v =µ/ρ to write the equationmore compactly

In the same way, the viscous forces in the x1 direction due to shear in the otherdirections can be derived, yielding

and the viscous forces in the other directions read

Including the friction term into the Euler equations we end up with the so-calledNavier-Stokes equations:

u x

r

1 3

2 1 3 2

u x

u x

r

j

1

2 1 1 2

2 1 2 2

2 1 3 2

2 1 2

r j r j

2

2 2

2 3 2

u

x j

1 2

1

2 1 2

u

x j

2 1

2

2 2 2

1+ = − ∂

∂ + ∂∂

du dt

The same result could have been derived by starting directly from Equation (3.8)and again using ρu i as the quantity to be conserved When neglecting spatial varia-tions of ρ in the diffusion terms (which is generally always a good approximation),the Navier-Stokes equations directly result with the diffusivity in Equation (3.8)being the viscosity just derived As in the Euler equations the additional forces arethe pressure gradient and the gravitational acceleration.

It may be interesting to note that the physical process responsible for molecularfriction is the molecular diffusion of the fluid particles If a faster fluid layer movesabove a slower fluid layer, some of the particles from the slower fluid layer will diffuseinto the faster layer, slowing down the upper layer On the other hand, faster fluid particlesdiffusing into the fluid layer below will accelerate the slower fluid layer From both sides

of the layer this results in an effective friction, either slowing down or accelerating theother layer Therefore, the operator is also called the diffusion operator

(3.16)

where v T is the molecular diffusivity for temperature, a parameter that depends only

on the properties of the fluid When compared to Equation (3.8) this is the

conser-vation of T with the external source given above.

3.4.1.4 Conservation of Salt

In the oceans and coastal lagoons the water has a certain salt content that variesfrom nearly zero high up in the estuaries and rivers to values of about 10 psu inbrackish water and up to more than 30 psu in ocean waters In lagoons where

p s

T

x c

Q x

3

1