WAVES AND SEA-AIR FLUXES. CALM CONDITIONS

Surface waves are the most obvious feature of the ocean. They result from the density discontinuity (we have discontinuity of about 800:1 at the surface). When the discontinuity is perturbed, two forces act to return the sea surface to the static equilibrium:

Gravity – body force Surface tension – elastic force

)tanh(12

2 kdg

kgk w

Dispertion relation

d is depth, γw is surface tension of the water, normalized with

kc

Phase velocity: group velocity:

kC

Velocity ofthe energy propagation

Main measures of surface wind waves:

04 mSWH

h - height

p - periodBasic measures:

The zero-upcrossing period is the time between two successive upcrossings of the mean level by surface elevation.

The zero-upcrossing wave height is the difference between the maximum and minimum values of the surface elevation between adjacent upcrossings of the mean level.

The crest period is the time between two successive crests.

The crest-to-tough wave height is the difference in height between a crest and the following tough.

Statistical measure: Significant wave height is defined in terms of spectral moments as

where m0 is the zeroth moment of the spectrum which is

equal to the sea surface variance.

WAVES GROWTH AND PROPAGATION

Waves are generated by wind and have right after generation the phase velocities, smaller than wind has. They are short in wave length and small in amplitude. Their spectrum is represented by a smooth peak in the high frequency range. If wind acts for a certain time, waves start to propagate faster, become longer and higher. In general they tend to increase their speed up to the wind speed. When their speed achieves the values of the wind speed, they become no more dependent on wind and start to propagate as free waves.

Surface waves, traveling slower than wind (i.e. waves still under

the wind influence) are called wind sea (sea). Wind sea traveling at wind speed is a fully developed sea. Waves, traveling faster than wind are swell.

SWH h hw s ( ) /2 2 1 2

Sea height Swell height

Estimation of SWH from sea and swell heights:

Propagation measuresWave age:

a = CP / Vef ,

where CP is the deep water wave phase speed at spectral peak,

derived from an estimate of the wave period pw:

CP = (g/2) pw ,

where g is the gravitational acceleration, and Vef is component of

the wind in the wave direction:

Vef = V10 cos ,

where is the angle between wave and wind directions and V10 is

the wind speed at 10-m anemometer height and neutral stability.

It is suggested, that for a < 1 wave can be regarded as sea, while

for a > 1, they should be considered as swell.

Wave length: = CP pw wave slope: = h/

kkkjki

jkjjji

ikijii

Water surface

Actual wind vector near the water surface

Tangential stressNormal stress

Through the normal stress wind works “to push” waves ahead, providing:

Acceleration of wave motion Transferring kinetic energy

Mechanics of wind-wave interaction: normal stress induced by waves:

If wind acts over a longer distance (fetch) or during a longer time (duration), waves tend to grow up, become longer and travel faster.

Wind stream functionsWind, not disturbed by waves

nt Tangential stress Normal stress

Thus, wind stress over waves should be expressed as:

(17)

How to parameterize the normal stress induced by waves?Key parameter: wave age

u

Caa

z p ,105.5 7.240

Smith (1991):

aCdn 24.285.1103

Toba et al. (1990) : laboratoryexperiment :

*2*

0 ,025.0u

Caa

u

gz pAlternative wave age – scaling with friction velocity. However, wind speed is a much easier available

parameter, than u*. Estimates of a’ vary within the range

from 4-5 (very young sea) to 30-40 (fully developed sea).

Donelan (1982): Lake Ontario:

where is rms wave height

Geernaert et al. (1987): MARSEN experiment:

3/2)(012.0 aCdn

5.4,09.7ln,0

BA

L

HA

H

zB

P

S

S

Taylor and Yelland, 2001: consideration of the relationship between the wave height and wave length:

where Hs is significant wave height, Ls is the peak wavelength.

Summary of sea-state-dependent wind stress: In general it should exist due to normal pressure components; The effect ranges from nearly 0 to 30%, reasonable estimate is of about 10%; The role of swell is uncertain, up-to-date concern: no effect of swell.

Estimates of the effect of wave-induced stress:(Gulev and Hasse 1998, JPO)

Smith 1988Traditional

estimate

Smith 1991Wave-age-

based stress

Ratio Sea-state dependent vs traditional

JAN

JUL

Major effect: Midlatitudes, Winter season, Up to 20%

Albedo decreases with windspeed by approximately 10-20% within the range 0-20 m/s Clear sky albedo decreases strongly than that under the cloudy sky

The other impacts of waves on sea-air exchange

Surface albedo: There are two major mechanisms through which labedo can be affected by the wind waves:

2. However, this effect seems to be not the largest. Under strong storms foam patches work to increase albedo. Thus, the total effect is the slight increase of the albedo. Decreasing with surface roughness albedo has been observed only under small solar declinations and nearly complete cloud cover.

Theoretical results of Presendorfer and Mobley (1986):

1. Multiply scattering of the SW from the rough surface – generally should act to decrease albedo. The main role belongs to capillary waves and not to developed seas.

Observational results, based on 32000 direct measurements worldwide:

Impact of the wave breaking on evapration (non-turbulent mechanisms of the water transfer from

the ocean to the atmosphere)

Wave breaking results in the generation of foam, which is normally represented by the so-called white caps at sea surface. Moreover, wave breaking results into generation of water drops, spread in the surface atmospheric layer. These drops are characterized by different from the sea surface characteristics (temperature, heat capacity, surface tension). Thus, the conditions of evaporation of these drops are quite different from the evaporation from the surface. Normally, they evaporate easier. Which part of the sea surface is covered by the white caps:

Measurements based on photos

Monahan et al (1982):

W(%)=3.8410-6u3.4

Normally drops do not “jump” too high (10-15 cm)– their further evaporation depends on how long they “live” in the surface layerbefore they are dumping back.How do the drops behave? How long they are living in the surface layer?

Lagrangian simulation ofdroplets at 12 cm height

Wind is 5-12 m/s. Life time: up to 1 sec

How to account for what is going on with the droplet, while it is flying in air?

Typical approach: modeling of the thermodynamics of droplets with field observations and laboratory experiments.

Architecture of a typical model (Bortkovsky 1982 “Sea-air exchange in storm conditions”):

System of equations of the droplet thermodynamics; Derivation of the droplet coordinates (2D) at every step from the droplet size and wind conditions; Estimation of the droplet freshening and surface tension; Estimation of the droplet temperature and heat capacity; Estimation of the droplet size; Computation of the heat and moisture loss by the droplet.

V

Ce, Ct

1

15-20 m/s

uqqCLQ

uCCQ

uC

zqq

ztph

zd

)(

,)(

,

0

0

2

What happens if U=0, even if temperature and humidity gradients are quite high?

=0

Q = 0 ???!!!

0

Calm conditions

Where we are with respect to the TKE?

MB

Dt

TKED

TC

gQB

z

uM

p

h

3*

When u* approaches zero,

the buoyancy flux does not!0

3*

h

p

gQ

uTCL

These are the called “free convection conditions”, where the heat and mass transfer is fully driven by the buoyant production. This case can be easily studied in laboratory experiments.

Paradox of bulk formulae:

Production of TKE by buoyancy (+/-)

(reversible)

Mechanicalproduction of TKE(normally positive)

Design of special experiment (Golitcyn and Grachov 1984)

1. Fully dark room 4x4x3 m;2. Tank of water, d=30 cm, h=1m, 3. One open surface, the walls are inconductive for heat;4. Temperature is measured at many levels;5. Air temperature is measured;6. The level of water is measured.

Temperature parameter:

])0(/[])([ awaw TTTtT t

A 13/1

Tw(0) – initial temperature, - scaling constant for time

LCmqTBo

m

BogkqLBQ

Bo

mgkTCAQ

p

qe

Tph

/),/()(622.0

,1)(

,1)(

4/123/4

4/123/4

is the thermal expansion of air, 0.61 is the analog of humid contraction, is kinematic viscosity of air, g is gravitational acceleration, A=0.144, B=0.159 are empirical constants.

Field measurements underV<3 m/s

Hasse, 1971: experimental measurements in the surface layer during day time and night time – dependence of the SST “skin-bulk” deviation from heat flux and wind speed.

Under calm and low winds absorption of the solar radiation occurs on a larger scale (wave length) than sensible and latent heat transfer (molecular scale). This leads to the fact that very thin upper layer of the water is in a lesser degree affected by SW, but, nevertheless is cooled by sensible a latent heat. This is surface cold skin layer. Its temperature has to be taken for estimation of fluxes and not the bulk temperature.

U

SWC

U

QCTT h

w 210

Variations of empirical coefficients with the reference depth:

Depth, m 0.25 0.5 1.0 2.5 5.0 10

C1 9.4 9,6 9.9 10.3 10.5 10.7

C2 1.61 1.75 1.90 2.14 2.32 2.52

Surface cool skin

/helios/u2/gulev/handout:

gra.for - Golitcyn and Grachov free convection scheme (1984)

SST=10C, Ta=8C, ez=9mb, SST=10C, Ta=12C, ez=11mb,