Sensible heat flux Latent heat fluxRadiation

Ground heat flux

Surface Energy Budget

The exchanges of heat, moisture and momentum between the air and the ground surface depend on the state of the atmosphere close to the surface. Since the air is a fluid, some basic notions of turbulence are needed to understand atmospheric motions close to the surface.

Viscous fluids

The viscosity causes an irreversible transfer of momentum from the points where the velocity is large to those where it is small. It is an internal resistance of fluid to deformation

The momentum flux density (also called shearing stress) due to viscosity is proportional to the gradient of the velocities.

z

u

In 1D

is called the dynamic viscosity and it is a property of the fluid.

Generalizing in three dimensions

i

j

j

iij x

u

x

u

Only for people interested in

the mathematical approach

tensor stress viscous called is ij

direction. -ythe invelocity of gradient

by induced stressshearing the of component-xthe is xy

.z,y,xj,i

So the units of are kg m-1 s-1

The ratio v=is called kinematic viscosity (units m2 s-1).

is a momentum flux density, so its dimensions should be those of momentum (M L T-1), per unit time (T-1) , per unit surface (L-2).As a consequence, the dimensions of are

1111

121

11121

TMLLLT

TLMLT

LLTTLMLT

L=lengthT=timeM=mass

(kg m-1 s-1) (m2 s-1)

Water 10-3 1. 10-6

Air 1.8 10-5 1.5 10-5

Alcohol 1.8 10-3 2.2 10-6

Glycerine 0.85 10-1 6.8 10-4

Mercury 1.56 10-3 1.2 10-7

Fluid particle of a viscous fluid adhere to solid surface. If the surface is at rest, the fluid motion right at the surface must vanish (No slip boundary condition).

Viscosity dissipates kinetic energy of the fluid motion. Kinetic energy is converted to heat.

To maintain the motion, the energy has to be continuously supplied externally, or converted from potential energy, which exist in the form of pressure and density gradients in the flow.

Osborne Reynolds (English, 1842-1912):“The internal motion of water assumes one or other of two broadly distinguishable forms – either the elements of the fluid follow one another along lines of motion which lead in the most direct manner to their destination, or they eddy about in sinuous paths the most indirect possible”

“Laminar”

“Turbulent”

In which conditions the flow is laminar and in which turbulent ?

Reynolds made an experiment the 22nd of February 1880 (at 2 pm).

He varied the speed of the flow, the density of the water (by varying the water temperature), and the diameter of the pipe. He used colorants in the water to detect the transition from Laminar to Turbulent.

He found that when

viscositykinematic

pipethe of diameter D

flowthe of speedU

~UD

20001900

The flow becomes turbulent

More in general, the ratio

with L typical length scale of the flow,is called “Reynolds number”.

ULRe

The critical value of the transition from laminar to turbulent flow changes with the type of flow

Reynolds number is very important in fluid mechanics

For Low Reynolds numbers the flow is Laminar

The flow behaves in two different ways for low and high Reynolds numbers.

A laminar flow is characterized by smooth, orderly and slow motions. Streamlines are parallel and adjacent layers (laminae) of fluid slide past each other with little mixing and transfer (only at molecular scale) of properties across the layers. A small perturbation does not increase with time. The flow is regular and predictable.

For high Reynolds numbers the flow is turbulent

Turbulent flows are highly irregular, three-dimensional, rotational, and very diffusive and dissipative. A small perturbation increases with time.

They cannot be predicted exactly as function of time and space. Only statistical averaged variables can be predicted.

m

F

Dt

uDamF

Newton’s second Law

For a fluid parcel

It links the acceleration of a fluid parcel, with the forces acting on the parcel.

Since momentum is conserved, you can think at the Newton’s second law as a budget equation. If the sum (in vector sense) of all the forces acting on a parcel is different than zero, there is a change in momentum.

Physical meaning of the Reynolds number

zyxx

P

Dt

Du xzxyxx11

For example, for u (x component) of the wind vector (but similar reasoning can be done for the others components)

Dt

DuTerm of Inertia. If the others terms are zero, the air parcel keeps moving at the same speed

x

P

1 Pressure forces (we’re not interested at this moment)

zyxxzxyxx1

Viscous forces. This can be seen also as budget (input-output) of momentum fluxes induced by viscosity

We focus only on the inertia and viscous terms

L

U

T

U

Dt

Du 2

If U is a characteristics velocity of the flow, and L a characteristic length, a characteristic time can be deduced from U=L/T

Inertial term

2

1

L

U

zyxxzxyxx

Viscous term

The ratio between the inertial and the viscous term give the relative importance of one respect to the other

ReUL

LUL

U

2

2

This is the Reynolds number

A high Reynolds number means that the inertial terms in the equation of motion are far greater than the viscous terms. However, viscosity cannot be neglected, because of the no-slip boundary condition at the interface.

The value of the critical Reynolds number (for the transition from laminar to turbulent flows) varies a lot from one case to the other (between 103 and 105). However in the atmosphere typical values are between 106 and 109.

Atmosphere is turbulent

Moreover, in the generation or damping of turbulence in the atmosphere a very important role is played by the buoyancy effects (we will see later on).

Properties of turbulence

Irregularity or randomness

Highly sensitive to small perturbations (changes in initial and boundary conditions). Unpredictable. Only statistical description can be used.

Three dimensionality and rotationality

The velocity field in any turbulent flow is three-dimensional, and highly rotational.

Diffusivity or ability to mix properties

Very efficient in diffuse momentum, heat and mass. “Macroscale” diffusivity of turbulence is usually many order of magnitude larger than the molecular diffusivity.

This is one of the most important properties concerning the applications. It is largely responsible of the dispersion of pollutants

Multiplicity of scales of motion and dissipativeness

Turbulent flows are characterized by a wide range of scales. The energy is transferred from the mean flow, to the larger eddies. There is a continuous transfer of energy from the largest to the smallest eddies. Viscous dissipation of energy occurs in the smallest eddies. In order to maintain the turbulent motion energy

must be supplied continuously.

Energy

dissipated

Mean and fluctuating variables

time

Win

d ve

loci

ty

Only averaged values of turbulent flows are predictable. Instantaneous values are random.

It is useful to split a variable in “mean” and “fluctuating” part

www

vvv

uuu

For example for the three wind components (u_> x, or West-East direction, v-> y or South-North direction, w _> z or bottom-up direction)

partg fluctuatinthe are w,v,u

part meanthe are w,v,u

values usistantaneathe are w v,u,

where

In the analysis of observations the most common mean or average used is the time average. For a generic variable f

T

dt)t(fT

1f

0

For micrometeorological observations, the averaging time T ranges between 103 and 104.

Reynolds decomposition

In modeling, more used are spatial averages

xyzV with

dz,dy,dx)x,y,x(fV

1f

zz

z

yy

y

xx

x

Average value over a grid cell

In some cases a spatial and time average is also performed.

z

x

y

In micro- and meso- scale simulations, x and y range between 102 and 103. z near the surface is between 10 and 102

In laboratory and some modeling and theoretical studies the ensemble average is used. This is an arithmetical average over a very large (approaching infinite) number of realization of a variable, obtained by repeating the experiment over and over again under the same general conditions.

nsrealizatio of numberN

fN

1f

N

ii

1

Nearly impossible to do in real atmospheric conditions

Theoretically the three averages are equal only for homogeneous and stationary flows. These conditions are very difficult to satisfy in micrometeorological applications. However, very often it is expected that an approximate correspondence between the results of the three averages exists.

fff

So, by definition

It is assumed that the average operator is such that the average of the fluctuations is zero.

fff 0

gfgf

gfgfgfgf

ggfffg

For the product of two variables

Co-variances or turbulent fluxes

Let consider a quantity c (per unit volume, ex: Mass per unit volume= density). If it is only transported, its flux density is a function of the wind speed

ucF

is )ms,st(density lux fthe So

stcu

is ssurface the

throughpassing c quantity the

of amountthe t,time the In

211

Splitting in mean and turbulent part and averaging

cucuccuuucF

cu The term

Is the the covariance, and represent the turbulent flux density

ut

s

For the momentum

juiuturbij

These are called Reynolds stress, and they are much larger in magnitude than the correspondent viscous stress

ixju

jxiuvisc

ij

The ratio is proportional to the Reynolds number

Revisc

ij

turbij

Turbulent mixing is much more efficient than viscous mixing

Variances are the simplest measure of the fluctuations

222222 w,v,u uvu

are also called standard deviations. The ratio of standard deviations over mean wind speed are called turbulence intensities.

wi,

vi,

ui w

wv

vu

u

In the atmosphere, turbulence intensities are less than 10% in the nocturnal boundary layer, 10-15% in a near-neutral surface layer, and greater than 15% in a unstable and convective boundary layers.

By combining the variances, it is possible to estimate the kinetic energy of the turbulent part, or Turbulent Kinetic Energy (TKE) per unit mass.

222

2

1wvuTKE

The kinetic energy of the flow is the sum of the Mean Kinetic Energy (MKE) and the Turbulent Kinetic Energy (TKE).

KE=MKE+TKE