ESCI 485 Air/sea Interaction Lesson 3 he Surface Layer References: Air-sea Interaction: Laws and Mechanisms, Csanady Structure of the Atmospheric Boundary Layer, Sorbjan HE PLANEARY BOUNDARY LAYER he atmospheric planetary boundary layer (PBL) is that region of the atmosphere in which turbulent fluxes are not negligible. Its depth can ary depending on the stability of the atmosphere. ο In deep conection the PBL can be of the same order as the entire troposphere. ο Usually it is of the order of 1 km or so. he PBL can be further broken down into three layers ο he mixed layer the upper 9% or so of the PBL in which eddies hae nearly mixed properties such as potential temperature, momentum, and moisture. ο he surface layer he lowest 1% or so of the PBL in which the turbulent fluxes dominate all other terms (Coriolis and PGF) in the momentum equations. ο he iscous sub-layer A ery thin layer (order of cm or less) right near the surface where iscous effects may be important. HE SURFACE LAYER In the surface layer the turbulent momentum flux dominates all other terms in the momentum equations. he surface layer extends upwards on the order of 1 m or so. he Reynolds fluxes (stresses) in the surface layer can be assumed constant with height, and hae the same magnitude as the interfacial stress (the stress between the air and water). In the surface layer, the eddy length scale is proportional to the height, z, aboe the surface.
Based on the similarity principle, the following dimensionless group is formed from the friction elocity, eddy length scale, and the mean wind shear where B is a dimensionless constant. Integrating Eq. (1) gies a logarithmic wind profile here B and C are constants. z du B u * dz, (1) U ( z) Bu *ln z + C () If z is defined as the leel where the mean wind speed is zero (called the roughness length), then C has the alue of C Bu *ln z (3) and we hae ( ) U ( z) Bu *ln z z. (4) he constant B is usually written (for some obscure reason) as B 1 κ (5) where κ is the on Karman constant, and has an empirical alue of.4. hus, the log-wind profile is normally written as ( κ ) ( ) U( z) u * ln z z. (6) Another way of writing the log-wind profile is to use an arbitrary reference leel, z r, and to call the elocity at this reference leel C. hen the log-wind profile is ( κ ) ( ) U ( z) u * ln z r + C, (7) where C and r are empirical constants that are found by obseration. WIND WAVES A steady wind blowing oer water will generate wind waes. Up to a point, the longer the wind blows, the higher the waes will become. After a certain period of time the wae height will be steady.
If the wind is blowing offshore, wae height will increase downwind up to a point, after which the wae height will be steady. (he distance oer which the wind is blowing is called the fetch.) Wae height is defined as the ertical distance from a trough to a crest (twice the amplitude). he significant wae height is defined to be the aerage height of the highest 33% of the waes, and is denoted at H 1/3. he restoring force for the longer waes is graity, so they are called graity waes. he restoring force for the ery shortest waes (λ ~ cm) is surface tension, and they are called capillary waes. We would expect that the significant wae height would depend only on the wind speed and graity. herefore, using dimensional analysis, we expect that H u g, (8) 1 3 * so that the combination u* /g can be thought of as a characteristic wae height scale. he phase speed C p is the speed at which an indiidual trough or crest adances. he group elocity, C g, is the speed at which the energy of the wae propagates. ο he group elocity is not usually the same as the phase speed. herefore, the energy of the wae propagates at a elocity different than that of what your eye sees following an indiidual crest. We call such waes dispersie. For most water waes, the group elocity is less than the phase speed, so that indiidual crests seem to moe through the packet of wae energy. New crests seem to be created at the trailing edge, moe through the packet of energy, and disappear at the leading edge. For ery long waelengths, (i.e., the waelength is larger than the depth of the water) the phase speed and group elocity are the same, and the waes are said to be non-dispersie. For ery short waelengths (capillary waes), where surface tension is the restoring force, the group elocity is actually greater than the phase speed. Waes enhance momentum transfer in seeral ways 3
ο he waes increase the roughness of the surface, proiding more interaction with the air. ο he wae faces allow horizontal pressure to act on the water. ο As the waes break, the turbulent motion mixes momentum deeper into the water. Eidence suggests that it is the shortest waes that are most efficient and important for air-sea momentum transfer, because they are steep and prone to breaking. CHARNOCK S LAW Obserations by Charnock in 1955 oer a reseroir established that the parameter r in Eq. (7) is related to the friction elocity and graity ia r u * g. his makes sense, since this is also a wae height scale, and we would expect wae height to factor into Eq. (7) somehow. Using Eq. (9) in Eq. (7) yields u * g z U ( z) ln C +. (1) κ u * Charnock also estimated that the constant C has a alue of about 1.5. Equation (1) is known as Charnock s Law. Since u* /g is a characteristic wae height scale (see preious section), Charnock s law says that the wind speed at a gien leel will decrease as wae height increases. his makes sense, since a higher wae height indicates a rougher surface. It is customary to use a height of z h 1 m when measuring and reporting wind data. herefore, Charnock s law becomes u * gh U ( h) ln C +. (11) κ u * (9) 4
DRAG COEFFICIEN What we would ultimately like is to somehow parameterize the friction elocity in terms of something we can easily measure, such as the wind as a certain height aboe the water, U(h). We can do this by rewriting Charnock s law as u [ U ( h) ] * C D h 1 gh ln + C κ u * where C D is the drag coefficient, and is [ U ( )], (1) 1 gh C D ln + C. (13) κ u * Notice that the drag coefficient itself is a function of u*. In reality, the drag coefficient is determined empirically, rather than through the use of the aboe formula, and is a function of wind speed, U(h). HE NON-NEURAL SURFACE LAYER Charnock s law assumes that the properties of turbulence in the surface layer depend only on friction elocity and height, u* and z, and that the wae height only depends on friction elocity and graity, u* and g. Implicit in these assumptions are that the surface layer is neutrally stable, meaning it is neither stable nor unstable. If the surface layer is stable, then ertical mixing is inhibited. his would be expected to reduce momentum transfer, and alter the mean wind profile by decreasing the winds at the lowest leels. If the surface layer is unstable, then ertical mixing is enhanced, and turbulent momentum transfer would also be expected to be enhanced, increasing the winds at the lowest leels. We therefore need an additional parameter to characterize the stability of the surface layer, and this is proided by the buoyancy flux, B. he perturbation buoyancy is defined as b g ρ g ρ, (14) 5
where ρ is the departure of density and is the departure from the aerage irtual temperature [ (1 +.61q)]. hrough some tedious manipulation (shown below) we get (because q <<1) 1 ( 1 +.61q ) ( 1 +.61q ) 1 ( 1 +.61q)( 1.61q ) +.61q ( 1 +.61q.61q ) 1 ( 1 +.61q ) 1 +.61q so that the buoyancy becomes 1 b g +. 61gq. (15) Buoyancy accounts for both the effects of heat and moisture on the stability of the atmosphere. Warmer temperature and higher humidity increase the buoyancy. he buoyancy itself can be thought of as a property that can be transported. If buoyancy is being added to the surface layer, than turbulence will be enhanced. If buoyancy is taken from the surface layer, than turbulence will be decreased. here is a ertical flux of buoyancy at the interface between the air and water. he Reynolds-aeraged buoyancy flux at the interface between the water and the air is B ( w b ). (16) WIND PROFILE IN HE NON-NEURAL SURFACE LAYER o include the buoyancy flux as an additional parameter needed to characterize the flow in the non-neutral surface layer we define a new length scale, called the Obukho length (L), as 3 u * L. (17) κb ο he Obukho length can be interpreted physically as the height aboe the surface when buoyancy first dominates oer mechanical production of turbulence. ο he Obukho length is positie for a stable surface layer. 6
ο he Obukho length is infinite for a neutral surface layer, since in a neutral surface layer there is neer a height where buoyancy dominates. ο he Obukho length is negatie for an unstable surface layer, since buoyancy dominates all the time, so you would hae to go to some negatie (irtual) altitude below the water to reach a point where buoyancy dominates (physically ridiculous, but it keeps the mathematics consistent). he wind profile in the neutral surface layer depended upon the friction elocity, eddy length scale, and the mean wind shear, which gae us z u * du dz 1 κ. (18) In the non-neutral surface layer we must also include the effects of stability, which we expect to be a function of both height and the Obukho length. We therefore write z u * du dz 1 φ κ z L, (19) where φ is the stability function, and is a dimensionless function of z/l. he stability function is determined empirically. A commonly used form is z z ϕ 1 + β ; L L z ϕ 1 L z 1 ϕ L 4 ( 1 α z L) stable neutral ; unstable. he figure below shows a qualitatie graph of the stability parameter as a function of z/l. () 7
he stability function modifies the wind profile in the surface layer. ο For the stable surface layer the winds are decreased. ο For the unstable surface layer the winds are increased ο here is no change for a neutral layer, which remains logarithmic. he figure below shows qualitatiely the wind profiles in the stable, neutral, and unstable surface layers. 8
MEHODS OF DEERMINING HE MOMENUM FLUX Determining the momentum flux in the surface layer can be achieed through either direct measurement of the turbulent momentum components, or by indirect methods. Some of these methods are described below. Profile method ο Use wind measurements at different leels to find the log-wind profile. ο From log-wind profile the friction elocity and roughness length can be inferred. ο Once the friction elocity is known the momentum flux is found ia u* ( w u ) 1 ο his method is not particularly accurate, and if the surface layer is not neutral, then additional complications from the stability parameter arise. Direct obseration of u and w ο his method requires an aerage oer long period of time (~ 1 hr), during which the elocity itself might change. ο Instruments to measure elocity components must be aligned ery precisely. ο he mast or tower holding the instruments may interfere with wind flow and introduce errors. Dissipation method ο Determine the energy spectrum of turbulence using Fourier transform. ο Use the the energy spectrum in the inertial subrange to calculate the dissipateion rate ε, from ϕ( k) a k 3 5 3 ε. (1) ο Assume that in the surface layer the dissipation rate is related to the friction elocity ia so that the friction elocity is 3 ε u * κ z, () u* ( κ zε ) 1 3. (3) 9