# Ocean near-surface boundary layer: processes and turbulence measurements

**REPORTS IN METEOROLOGY AND OCEANOGRAPHY **

**UNIVERSITY OF BERGEN, 1 - 2010 **

**Ocean near-surface boundary layer: **

** processes and turbulence measurements **

**M**

**OSTAFA **

**B**

**AKHODAY **

**P**

**ASKYABI**

## and

**I**

**LKER **

**F**

**ER**

Geophysical Institute

University of Bergen

## December, 2010

**«REPORTS IN METEOROLOGY AND OCEANOGRAPHY» **

**utgis av Geofysisk Institutt ved Universitetet I Bergen. **

**Formålet med rapportserien er å publisere arbeider av personer som er tilknyttet **

**avdelingen. **

**Redaksjonsutvalg: **

**Peter M. Haugan, Frank Cleveland, Arvid Skartveit og Endre Skaar. **

**Redaksjonens adresse er : «Reports in Meteorology and Oceanography», **

**Geophysical Institute. **

**Allégaten 70 **

**N-5007 Bergen, Norway **

**RAPPORT NR: 1- 2010 **

**ISBN 82-8116-016-0 **

2

**CONTENT **

**1.** **INTRODUCTION ................................................................................................2**

**2.** **NEAR SURFACE BOUNDARY LAYER AND TURBULENCE MIXING .............3**

**2.1.**

**Structure of Upper Ocean Turbulence ........................................................................................ 5**

**2.2.**

**Governing Equations......................................................................................................................... 6**

2.2.1.

Turbulent Kinetic Energy and Temperature Variance .................................................................. 6

2.2.2.

Relevant Length Scales..................................................................................................................... 8

2.2.3.

Relevant non-dimensional numbers................................................................................................ 8

2.2.4.

Turbulence Spectra ........................................................................................................................... 9

**2.3.**

**Atmospheric Forcing ....................................................................................................................... 10**

2.3.1.

Wind Stress....................................................................................................................................... 10

2.3.2.

Bulk Heat Fluxes .............................................................................................................................. 11

2.3.3.

Solar Radiation Fluxes..................................................................................................................... 12

2.3.4.

Precipitation...................................................................................................................................... 13

2.3.5.

Surface Buoyancy Flux.................................................................................................................... 13

2.3.6.

Gas Fluxes......................................................................................................................................... 14

**3.** **OCEAN TURBULENCE MEASUREMENT.........................................................14**

**3.1.**

**Measurement systems ................................................................................................................... 14**

**3.2.**

**Sensor Features................................................................................................................................ 21**

3.2.1.

Temperature..................................................................................................................................... 21

3.2.2.

Conductivity...................................................................................................................................... 21

3.2.3.

Shear ................................................................................................................................................. 22

**3.3.**

**Parameter Estimation..................................................................................................................... 23**

3.3.1.

Turbulence Dissipation Rate .......................................................................................................... 23

3.3.2.

Vertical Eddy Diffusivity for Mass .................................................................................................. 24

3.3.3.

Dissipation Rate of Temperature Variance and Eddy Diffusivity for Heat............................... 24

3.3.4.

Vertical Overturns............................................................................................................................ 25

**3.4.**

**Measurement Limitations and Platform Motion ................................................................... 26**

3.4.1.

Platform Motion and Vibration Correction Procedures ............................................................... 26

3.4.2.

Despiking and Denoising ................................................................................................................ 30

**4.** **TURBULENCE NUMERICAL MODELING .......................................................30**

**4.1.**

**Direct Numerical Simulation........................................................................................................ 30**

**4.2.**

**Large Eddy Simulation ................................................................................................................... 30**

**4.3.**

**Statistical Turbulence Modeling ................................................................................................. 31**

**4.4.**

**Empirical Turbulence Modeling................................................................................................... 31**

**5.** **SUMMARY ........................................................................................................31**

**6.** **REFERENCES....................................................................................................32**

1

**Abstract **

E xchange of mass, momentum and heat by turbulent processes is crucial in the air-sea interactions. In this report, state-of-the-art of turbulent kinetic energy dynamics in the upper ocean boundary layer is reviewed. Particular attention is given to the structure of the upper mixed layer, main turbulent processes in this layer, different methodology of near surface turbulence measurements, and complementary turbulence numerical models.

**1. Introduction **

The complex air-sea interface sets the boundary conditions for physical and biogeochemical processes and plays a key role in the global atmosphere-ocean heat balance.

Fluxes across this interface influence the weather, climate and spatial distribution and evolution of greenhouse gases. The upper water column below the air-sea interface comprises of an actively entraining upper ocean mixed layer, the upper 10-50 m well-mixed in temperature and salinity, bounded at its base by a sharp density discontinuity separating the layer from a stable, essentially non-turbulent pycnocline. The thermal and mechanical energy received from the atmosphere in the upper layer not only control the local dynamics, but the layer itself modulates the flux of this energy to the deeper water masses (Garwood 1976).

This mixing process plays a major role for the dynamics of marine ecosystems. It provides food web, light and nutrients for phytoplankton population (Gargett 1997). The first classical study of the surface mixed layer and shelf sea waters was done by Sverdrup (1953) about the development of spring phytoplankton blooms as a function of light supply and the depth of vertical mixing. Blooming can occur only when the depth of the mixed layer is less than a compensation depth at which gain and loss of the population balance each other.

Sverdrup’s theory thus necessitated a better understanding of the complex upper ocean dynamics, of erosion of the pycnocline, and especially of the turbulent mixing processes which control the exchange and variability of properties such as momentum, heat, mass and gases in the upper ocean. Furthermore, understanding turbulence and its role in ocean mixing is key in critical ocean processes (Nihoul 1980), and needed even in the studies of ocean circulations and the coupled ocean-atmosphere climate.

Upper ocean turbulence has a broad spectrum of time and spatial scales, forced by surface fluxes of buoyancy, momentum and surface waves. The buoyancy loss at the surface due to surface cooling and evaporation (and due to ice freezing at high latitudes, which is not considered in this report) results in convection due to unstable surface density stratification.

The overturning eddies will lead to entrainment of water from below the pycnocline. The mean shear due to loss of mean flow kinetic energy generated by wind stress over sea surface

(and its associated currents) can lead to instability of Kelvin–Helmholtz billows with their axes oriented roughly at right angles to the shear (Li et al. 2005). The interaction between the wind-driven Stokes drift of surface waves and the mean shear generates counter-rotating vortices aligned with the wind direction, so-called the Langmuir circulation (Thorpe 2004,

2007). Langmuir circulation is a means of efficient energy transfer from the surface wave energy to turbulence mixing (Grant and Belcher 2009; Teixeira and Belcher 2010). Therefore, the upper ocean turbulence and associated mixing are sensitive to and dependent on the properties of the surface gravity waves (Ardhuin and Jenkins 2006). Deeper in the water column , internal wave breaking, dynamical Kelvin-Helmholtz instabilities at the pycnocline

2

and local frictional drag against the topography are other typical trigger mechanisms which contribute significantly to the vertical redistribution of both momentum and scalars in the ocean interior (Thorpe 2004, 2007).

Understanding of the aforementioned processes at the sea surface and the structure of the boundary layers requires careful observations and measurements of turbulence mixing. Direct measurements of turbulence are very limited and difficult. Surface wave orbital velocity fluctuations are often about 1 m s

-1

, whereas the velocity scale of turbulent velocity fluctuations in the near surface is about 1 cm s

-1

, illustrating the measurement complexity.

Thus, the surface wave orbital velocity fluctuations are two orders of magnitude larger than the turbulence velocity fluctuations, so they represent a large signal that must be separated from the less energetic turbulent signal in order to compute turbulent variances, co-variances, and dissipation rates (Soloviev et al. 1999). Indirect methods are often used to study ocean turbulence mixing by using appropriate measurement techniques, sensors, and theoretical approximations (Burchard et al. 2008b). A useful parameter for characterizing the turbulence field is the viscous dissipation rate of turbulent kinetic energy per unit mass,

. Different approaches to measure ocean turbulence include profiling or towed microstructure measurements, time series from fixed sensors, as well as particle image velocitimeter (Section

3). The vertical fluxes of heat and buoyancy are usually obtained by assuming a balance between turbulent production, buoyancy flux and dissipation (Osborn 1980) and a constant mixing efficiency. For the past 40 years, the standard technique of measuring turbulent quantities in the ocean has been the use of vertical microstructure profilers, pioneered by

Osborn (1974) and Gregg et al. (1973).

In this report, we review the current understanding of the processes at the sea surface and in the near surface boundary layer, including theory and observational techniques. Recently, numerical modeling has been extensively applied to analyze the dynamics of turbulent mixing. For completeness, a brief description of typical numerical tools for turbulent mixed layer modeling (direct numerical simulation, large eddy simulation, statistical and empirical models) is also presented. The report is organized as follows. Near surface structure and dynamics with the state-of-the-art in turbulence mixing are presented in Section 2.1. The governing equations and relevant non-dimensional numbers are reviewed in Section 2.2.

Important components of the atmospheric forcing are summarized in Section 2.3. Turbulence measurement and instrumentations are reviewed in Section 3. Subsequently, in Section 4 numerical models of turbulence in the mixed layer are introduced briefly, followed by the concluding remarks.

**2. Near Surface Boundary Layer and Turbulence Mixing **

The near surface boundary layer is the upper part of ocean that is dominantly (directly or indirectly) influenced by the surface fluxes such as heat, moisture, momentum, wind stress and surface waves (Anis and Moum 1995). It comprises the interface between the ocean and atmosphere where many processes occur, such as breaking waves (upper few meters), molecular transport (upper few millimeters), diurnal and seasonal convection, and Langmuir circulation (Soloviev and Lukas 2006). Turbulence acts to remove the vertical gradients of oceanic tracers (temperature, salinity and density) above the pycnocline leading to a quasihomogenous layer: the upper ocean mixed layer. While the profiles of turbulent dissipation in the upper layer define the mixing layer depth, the mixed layer depth and its evolution is

3

shaped by an integral of mixing events (Brainerd and Gregg 1995; Fer and Sundfjord 2007).

At instances when the mixing layer penetrates deeper than the mixed layer, turbulent motions work against the stratification and a part of the energy input is used to increase the potential energy of the water column. At the base of the mixed layer is a transitional region with strong density gradient where the density increases with depth (pycnocline). Typically this layer corresponds to the thermocline where the temperature decreases with depth. Generation and propagation of internal waves and formation of strong velocity gradient can occur in the pycnocline (not covered in this report).

The near surface boundary layer, including the mixed layer, is a key component in the study of climate, biological productivity and marine pollution (Thorpe 2007). Transfer of momentum, heat, gases across the ocean surface, and the turbulent motions within the relatively uniform mixed layer of the ocean are largely controlled by waves, especially breaking waves (Thorpe 2004, 2007), local source of shear and convective instabilities or nonlocal source by convection and advection (Soloviev and Lukas 2003b, 2006). Figure 1 illustrates some of the processes in the upper ocean. Diurnal cycle of solar radiation (heating) reduces the density of the upper water column, thereby suppressing the near surface turbulence. Cooling of the upper surface due to evaporation and long wave radiation leads to convection (positive buoyancy flux) into the interior of water column and develop turbulence.

Breaking waves are the main source of turbulent kinetic energy (TKE) in the top meters of the ocean. This production of TKE is believed to occur mostly through the large shear at the forward face of breaking waves (Ardhuin and Jenkins 2006). Ocean-atmosphere interface is a two phase boundary with air bubbles in water and sea spray in air. The near surface processes are sensitive to the appropriate conditions in this two phase boundary. Bubbles from breaking waves change the density stratification and affect the near-surface thermohaline stratification.

Understanding of the small-scale turbulence depends on understanding of the multiple interactions between the turbulence, currents, and surface waves. In this section, a brief description of the upper ocean structure is given in section 2.1, followed by the governing equations (section 2.2). Finally, basic components of the atmospheric surface forcing are presented in section 2.3.

**Figure 1. ** An illustrated view of the most important processes shaping the upper ocean boundary layer. The scales of the drawing are arbitrary ( http://www.us-solas.org:8080/Plone).

4

**2.1. Structure of Upper Ocean Turbulence **

Wind waves supply energy for turbulence through breaking and impact strongly the airsea system on all scales. Due to breaking and vortex instabilities at the ocean surface, waves release TKE and generate a source of turbulence at the ocean surface. Thus, turbulent vertical mixing is intensified within the near surface layer (Qun et al. 2007). Furthermore, exchange of atmospheric gases such as oxygen and carbon dioxide occurs by diffusion across the sea surface and across the surfaces of bubbles created by the breaking waves (Garrett 1996).

Phenomena such as surface waves (especially breaking waves), bubbles, shear instabilities, surface heating and cooling influence TKE and its dissipation rate in the water column.

Ignoring stratification and rotation effects, the upper ocean turbulent boundary layer can be conveniently divided into four sublayers as illustrated in figure 2 (Soloviev and Lukas 2006):

1- the wave-stirred layer where the TKE production by wave breaking significantly exceeds the mean shear effect, and the turbulent diffusion of the wave kinetic energy dominates in the range of depths where the wave motion continues to be vigorous,

2- the turbulent diffusion layer where the turbulent diffusion of TKE exceeds the wave

(as well as the mean shear) effect in the TKE budget,

3- the wall layer where the mean shear production of turbulent energy dominates, and

4- the viscous sublayer. Viscous sublayers develop at both the water- and air-side of the air-sea interface due to the suppression of the normal component of turbulent velocity fluctuations near the density interface. This sublayer is controlled by the tangential wind stress.

The most significant characteristic of the ocean mixed layer, in contrast to the atmospheric boundary layer, is the presence of wave breaking and the Langmuir circulation at the free surface. Breaking of surface waves generates large amounts of small-scale turbulence near the sea surface, and the interaction between the wind-driven surface shear and the Stokes drift of surface waves generates Langmuir circulations that are large circulation cells aligned in the wind direction (Noh and Min 2004). Measurements revealed that in the presence of wave breaking, the dissipation rate of TKE near the sea surface is several orders of magnitude larger than that expected from the law of the wall (section 2.3.1) (Anis and Moum 1995).

Furthermore, wave breaking increases the roughness length scale, *z*

0

, at the sea surface to 0.1–

8 m, much larger than that of the atmospheric boundary layer. The first numerical attempt to model the wave-enhanced turbulence near the surface was done by Craig and Banner (1994), who used the level 2.5 turbulence closure scheme of Mellor and Yamada (1982) with a simple model of the oceanic boundary layer to successfully reproduce the observations from a number of datasets. Their model predicts significantly enhanced surface turbulence relative to the law of the wall (Thorpe 2007). They suggest that *z*

0 in the oceanic boundary layer is of the order of the surface wave amplitude, much larger than that in the atmospheric side of the air– sea interface. Craig (1996) notes that *z*

0 is a significant scaling parameter for the dynamics in the boundary layer affected by waves, but that the dissipation measurements in the ocean or lakes are not yet precise enough to enable an accurate estimation of its magnitude. Clearly, *z*

0 is an important length scale but it is as yet hard to quantify (Stacey 1999).

The large-scale Langmuir circulations as a source of convective motions in the ocean surface layer are another result of wind-induced flow in presence of surface gravity waves (Li,

5

2009). Skyllingstad and Denbo (1995) used a large eddy simulation model to simulate the

Langmuir circulation and convection in the surface mixed layer and carried out numerical experiments with different combinations of wind stress, wave forcing, and convective forcing.

In these simulations the Langmuir circulations were found to dominate the vertical mixing, leading to enhanced dissipation rates that were consistent with observations of Lombardo and

Gregg (1989). McWilliams et. al. (1997) introduced the turbulent Langmuir number,

*L*

*at*

5

*u*

*****

*u s*

0

0

**.**

5

, where

*u*

*

is the surface friction velocity in the water and *u * is the surface

*s*

0

Stokes drift. *L * is now recognized as an important parameter describing the Langmuir

*at*

turbulence (Grant and Belcher 2009). Furthermore, recent work suggests that high-frequency internal waves generated by Langmuir circulation over stratified water may be an important source of turbulent mixing below the surface mixed layer (Polton et al. 2008).

**Figure 2. **Diagram of the upper ocean turbulent boundary layer. Here *h*

W-S

is the wave stirred layer depth, and *h*

TD

is the turbulent diffusion layer depth (Soloviev and Lukas 2006).

**2.2. Governing Equations **

The well-known Navier-Stokes equations for the analysis of Newtonian fluid dynamics include conservation laws for mass (continuity equation), momentum, heat, energy, salt, humidity, and the equation of state. Presentation and derivation of the basic equations can be found in e.g., Tennekes and Lumley (1972), Gill (1982) and Pond and Pickard (1983). The next section presents the budget equations for TKE and the temperature variance. To provide a framework for near surface turbulence, we then present a brief summary of different length scales, characteristic non-dimensional numbers, parameters and universal spectral forms.

**2.2.1. **

*Turbulent Kinetic Energy and Temperature Variance *

The TKE budget equation is

6

*t*

*U j*

*x j*

*e i*

*x j*

1

2

*U u i j*

*u e j i*

*A*

1

*p u j*

2

*i ij*

*i j*

*B*

*U x j i*

*i*

*C*

*D*

*ij ij*

(1) where the primes and the overbar denote fluctuations and averaging, respectively. Here

*U*

is

*j*

the mean field velocity vector,

is the kinematic viscosity, density,

*u*

1

,

*u*

2

,

*u*

3

is a constant reference

0

*b*

*g*

0

is the buoyancy anomaly, *g* is the gravitational acceleration,

is the velocity in the standard orthogonal coordinate system where directions

*x*

1

, *x*

2

and *x * are positive to the east, north and upwards, respectively. The strain rate tensor

3

*s* is given by

*ij s ij*

1

2

*u i*

*x j*

*u j*

*x i*

(2)

The products in equation (1) are taken over repeated suffices

*i*

,

*j*

1 , 2 , 3 . The first term in equation (1) expresses the local storage of variance or TKE. The other terms are

Term **A**: Transport terms containing energy transported by the mean flow and turbulence respectively (first and second terms), the pressure correlation term (third one) and the viscous transport term (last term).

Term **B**: Shear production term, the rate at which TKE is produced by the interaction of the Reynold’s stress,

*i*

, with the mean velocity shear.

Term **C**: Buoyancy flux (*B*). In the case of an unstable density stratification, the buoyancy flux is positive; it will be a TKE production term. In stable stratification, turbulent mixing induces a negative buoyancy flux by displacing the dense water upward, hence *B* becomes a sink in equation (1).

Term **D**: Dissipation rate,

. It is the loss of kinetic energy of turbulent motion per unit mass through viscosity to heat.

By Reynolds averaging of the heat equation, the temperature variance equation can be derived as

*j*

*x*

1

2

*T*

2

*t u*

*A*

*x*

*j*

*k*

*T*

*x j*

1

2

*T*

2

*B*

*x*

1

*T*

2

*u*

*j j*

2

*C*

2

*T*

*k*

*T*

*x*

*D*

*u*

*j*

*T*

*T*

*x j*

*E*

(3) where

*k*

*T*

is molecular diffusivity for heat, *T * is temperature fluctuation. The terms are

Term **A**: Storage term.

Term **B**: Molecular transport.

Term **C**: Turbulent transport.

Term **D**: Dissipation rate of temperature variance.

Term **E**: Gradient production.

7

**2.2.2. **

*Relevant Length Scales *

The following length scales are descriptive of mixing in a turbulent flow.

**Kolmogorov length scale**: It is defined as

1

**/**

*k*

*K*

with the Kolmogorov wavenumber

*k*

*K*

(cyclic units)

*k*

*K*

1

**/**

4

(4)

It represents the scale at which viscous forces equal the inertial forces, and represents the lower end of TKE at the dissipation subrange of the spectrum. The relation between a cyclic wavenumber, *k*, and a radian wavenumber,

*,* is *k* =

/2

.

**Batchelor length scale**: It is defined as

1

**/**

*k*

*B*

where

*k*

*B*

is the Batchelor wavenumber

*k*

*B*

*k*

*T*

2

1

**/**

4

(5) expressed in cyclic units, and

*k*

*T*

is molecular diffusivity for heat. It is the length scale at which temperature fluctuations disappear. An equivalent Batchelor length scale for salinity can be written by replacing the diffusivity of heat with the diffusivity of salt.

**Ozmidov length scale**: It is defined as the length scale at which buoyancy forces balance the inertial forces, expressed as

*L*

*O*

*N*

3

1 2

(6) where

*N*

is the buoyancy frequency,

*N*

2

*g*

*d*

*dz*

*d*

*dz adiabatic*

*g*

*d*

*dz*

where

is the potential density anomaly.

**Corrsin length scale**: It is the smallest scale at which eddies are strongly deformed by the mean shear, estimated as

*L*

*C*

*S*

3

1 2

(7) where

*S*

2

*du dz*

*dv dz*

2

, and *u *and *v * are the orthogonal components of the mean horizontal velocity. Corrsin length scale represents the upper limit of eddy sizes when the background shear is more restrictive than the background stratification.

**2.2.3. **

*Relevant non-dimensional numbers *

**Gradient Richardson number: **It is defined as the ratio of squared buoyancy frequency and squared mean shear** **

2

*Ri N S*

2

. (8)

**Flux Richardson number: **It is defined as the ratio of buoyancy production/consumption term in equation (1) (i.e., negative of the buoyancy flux,

*B*) to shear production term. For stable stratification, *Ri*

*f*

is the efficiency of

8

mixing. When the mean flow *u * is aligned with the *x*-axis, the flux Richardson number is given as

*Ri f*

*w*

*b*

*u*

*w*

*u*

*z*

(9)

**Mixing Efficiency factor: **It is given as

*R f*

1

*R f*

, (10) and a typical upper limit of 20% is often used (Moum 1996).

**Buoyancy Reynolds number: **It is a non-dimensional number which characterises the vigour of turbulence. It is also called the turbulence activity index, and is also indicative of the likelihood of local isotropy (see section 3.3.1). It is defined as** **

*I*

(11)

**2.2.4. **

*Turbulence Spectra *

Spectral analysis is a widely-used approach in turbulence studies since the wavenumber or frequency domain distribution of velocity, or scalar variance can be readily related to scale and evolution of eddies, and the rate of their dissipation. A theoretical form of the spectrum of TKE was hypothesized by Kolmogorov that covers the broad range of wavenumbers up to a cut-off wavenumber limited by the effect of viscosity (the Kolmogorov wavenumber, Eq. 4). According to Kolmogorov’s hypothesis, there exists an inertial subrange where the three-dimensional velocity wavenumber spectrum of a high Reynolds number flow has a universal form that depends only on the energy dissipation. Assuming isotropic turbulence, Kolmogorov’s spectrum is given by following functional relationship

*S*

*K*

*C*

2 3

*k*

5 3

(12) where

*k*

is the radian wavenumber and

*C*

0**.** 5 is the universal Kolmogorov constant. This simple relationship between energy dissipation and the wavenumber spectra allows the estimation of

using the inertial-dissipation method.

The Batchelor spectrum, here given for the temperature gradient, in one dimension can be written as (Dillon and Caldwell 1980)

*S*

*B*

1 2

*T k*

*B*

1

*k*

*T*

1

*f*

(13) where *q * is a universal constant,

*T*

is the temperature variance dissipation rate due to turbulence. Assuming isotropy

*T*

6

*k*

*T*

0

*S*

*B dk*

6

*k*

*T*

*T*

*z*

2

(14) where

*k*

*T*

is the molecular diffusivity for heat,

*T*

/

*z*

is the small scale vertical temperature gradient. In equation (13),

is a non-dimensional wavenumber equal to

*kk*

*B*

1 2

. The nondimensional shape of the spectrum is given by

9

*f*

*e*

2

2

0

*e x*

2

2

*dx*

(15)

The universal constant *q * has been estimated as

*q*

3 .

67

1 .

52 (Oakey 1982).

**2.3. Atmospheric Forcing **

**2.3.1. **

*Wind Stress *

The wind stress, or momentum flux, is a driving force for surface gravity waves and oceanic currents (Garrett 1996). This momentum exchange between air and sea is determined to a large extent by the aerodynamic roughness at the air–sea interface since it determines the turbulence level near the interface, and thereby the wind stress. Stress is nearly constant with height in the surface layer; the mechanisms, however, vary with height. In a wave boundary layer, much less than a wavelength in height, part of the stress is carried by the wave-coupled motion, while away from it nearly all the stress is due to the turbulent momentum flux. In a viscous sublayer, generally less than 1 mm thick, the turbulent momentum flux gives way to viscous shear stress (Smith et al. 1995). At the sea-air interface this stress is expressed as

*a u*

*

2 where

a

is the density of air and

*u*

*

is the friction velocity which can be thought of as a proxy for the surface wave. One of the difficulties in the turbulent near surface layer is the formulation of the aerodynamic roughness, *z *, and its relationship to

0

*u*

*

. On dimensional grounds

*z*

0

*u*

*

2

*g*

(16) where

0**.** 0185 is the Charnock constant (Toba and Kunish 1970; WOM 1998). The wind velocity develops a logarithmic form (Wu 1983)

*u*

*u*

*

ln

*z z*

0

(17) where

*u*

is the velocity profile, *z * is the height, and

0 .

4 is the von-Karman constant. A common reference level for turbulence in boundary layers in the absence of buoyancy forces is the constant stress layer scaling, also called the law of the wall (Tennekes and Lumley

1972). For a mixed layer of depth

*h*

, the dissipation rate,

, profile in the near surface layer can be divided into inner ( 0

*z*

0 .

1

*h*

) and outer

0**.** 1

*h*

*z*

*h*

parts. In the inner part, the dissipation profile is closely related to the wind stress production and decays with depth following

*u*

*

3

*z*

*z*

0

. (18)

The velocity profile given by equation (17) can be rewritten as (Taylor 1972)

*u*

*u*

*

ln

*z z*

0

*z*

0

(19)

10

Dissipation in the outer part is influenced by other environmental parameters such as rotation and stratification. However, both the inner and the outer parts are affected by the depth of the mixed layer which evolves in response to heat fluxes, wind stress and the Earth’s rotation

(Soloviev and Lukas 2006). It is common to express wind stress,

, through the following bulk transfer relation

*a*

*C*

*D*

where

*C*

*D*

is the drag coefficient and

*u*

is given in equation (17). If the boundary is in a state of neutral stratification, the drag coefficient can be expressed as

*C*

*D*

*u u*

*****

2

(20)

However, the boundary layer over the ocean is not necessarily neutral. A stability dependence is derived using the Monin-Obukhov similarity theory as

*C*

*D*

ln

*z z*

0

*z*

*L*

*MO*

1

(21)

The function

(Stull 1988) is given for stable and unstable conditions according to the

Monin-Obukhov length

*L*

*MO*

*u*

*****

3

*J b*

0

(22) where

0

*J * is the surface buoyancy flux. Further, the logarithmic velocity profile (Eq. 17), can

*b*

be revised as

*u*

*u*

*****

ln

*z z*

0

*z*

*L*

*MO*

At the air–water interface the energy input equals the sum of the surface stress,

, per unit mass times an effective phase speed,

*c eff*

, of waves acquiring energy from the wind and the input to the surface drift,

*u*

, i.e.,

*s*

*E in*

*c eff u*

*

2

*u s u*

*

2

(Wu 1975). The second term is generally small (Gemmrich 2010). In a fully developed wave field, the total energy dissipation,

*E*

*h*

*dz*

, balances the energy input, i.e.,

*E*

*E in*

. Thus, the dissipation rate plays an important role in the energy balance, especially if it is affected by breaking waves. However, it is typically difficult to measure the dissipation rate of TKE near the ocean surface, especially in the presence of wave-induced motions (Zhang and Chan

2003; Weller et al. 2004).

**2.3.2. **

*Bulk Heat Fluxes *

Turbulent and radiative processes exchange heat between the atmosphere and the sea, and together with evaporation and precipitation, modify air and water masses at the surface

11

(Smith et al. 1995). The turbulent fluxes of the sensible,

*Q*

*T*

, and latent,

*Q*

*E*

, heat and the net long wave radiation

*Q*

*L*

are defined by the standard bulk formulae

*Q*

*T*

*Q*

*E*

*Q*

*L*

*a c pa*

*a*

*L*

*V w*

*T*

*w*

*q*

*a c pa*

*a*

*L*

*V*

*C*

*T*

*C*

*E*

*U*

10

*U*

10

*q*

*T s s*

*T a q a*

(23)

*w*

*T s*

4

*w*

*E a*

where

*c pa*

is the specific heat capacity of air at constant pressure, and

*w*

**,**

*T*

and *q* are the fluctuations of vertical velocity, temperature and specific humidity, respectively.

*L*

*V*

is the specific latent heat of vaporization,

*C*

*T*

and

*C*

*E*

are the drag coefficients or bulk transfer coefficients for sensible and latent heat,

*T*

is the sea surface temperature,

*s*

*T*

is the air

*a*

temperature at 10 m height,

*q*

is the water vapor ratio (saturation specific humidity) at the

*s*

ocean surface level,

*q*

is the specific humidity of air at the standard level of 10 m above sea

*a*

surface,

5 67

10

8

Wm

2

K

4

is the Stefan-Boltzmann constant,

*w*

0 .

97 is the infrared emissivity of water, and

*E a*

is the long wave irradiance from sky. The drag coefficients in equation (23) can be calculated as (Rodriguez 2005)

*C*

*T*

1 /

*C*

*D*

2

ln

*z z*

0

1

*C*

*E*

*C*

1/ 2

*T*

ln

*z*

*z*

0

1 and

*C*

*D*

is given by equation (20) or (21) according to the stability conditions. Alternatively, empirical formulae can be used: *C*

*E*

= 1.5

10

-3

and *C*

*T*

= 1.1

10

-3

for unstable and 0.83

10

-3 for stable conditions (Soloviev and Lukas 2006). Furthermore, the aerodynamic roughness length includes a correction term to equation (16),

*z*

0

*g u*

*

2

*u*

*

(24)

Where

= 0.11. The second term in equation (24) has negligible influence for wind speeds above about 6 m s

-1

*. *

The value of *z*

0

in equation (24) is determined by an iterative process

(Smith et al. 1995).

**2.3.3. **

*Solar Radiation Fluxes *

The incoming solar radiation at the top of the atmosphere is determined by the solar elevation and also by the orbital variations in distance between the Earth and the Sun. In the atmosphere, clouds and aerosols reflect and absorb a fraction of the incoming radiation. The albedo (fraction of solar energy reflected at the surface) depends on solar elevation, sea state, ice cover, and cloud cover. The incoming energy is absorbed in the top few meters, depending on the clarity of the water which in turn depends on the plankton and sediment present in the water. The net long-wave radiation is usually, but not always, upward. The loss is nearly all from the top few mm and is responsible (together with the latent heat flux) for a “cool skin” that is often a few tenths, or even a degree, cooler than the bulk of the near-surface water

12

temperature. The upward component depends on the sea temperature and on the emissivity of sea water. The downward long-wave radiation from the overlying air is more complex, depending on profiles of air temperature and humidity and on the presence of fog and cloud.

The net long-wave radiation is given as

*Q*

*B*

4

*w*

*T a*

3

(

*T s*

*T a*

)

*w*

*T a*

4

( 0 .

39

0 .

05

*e*

*a*

)

*c f*

(25) where

*e*

is water vapor pressure,

*c f*

fractional cloud cover.

1

0 .

72

*C*

, and

*C*

is a cloud correction factor, or the

The short–wave net radiation based on latitude, longitude, time, fractional cloud cover and albedo is (Payne 1972)

*Q*

*S*

*Q s*

0

1

*s*

1

0 .

6

*C*

3

(26) where

*s*

is the mean albedo over a period of time, often a month, and

*Q s*

0

is the short-wave radiation reaching the surface under clear sky.

**2.3.4. **

*Precipitation *

The evaporation and precipitation imbalance is a major branch of the hydrological cycle. The measurement of precipitation at sea remains a problem. Data exist mainly from a few atolls and small islands (Smith et al. 1995). Large and variable flow distortion makes it extremely difficult to measure precipitation from moving ships, and there are no valid data from the Voluntary Observing Ships that otherwise provide the bulk of the marine meteorological data (Li 1998).

**2.3.5. **

*Surface Buoyancy Flux *

The surface buoyancy flux is given by

*J b*

0

*g*

*c*

*P*

*Q*

*J*

*S*

0

(27) where,

0

*J*

is the surface salinity flux due to evaporation-precipitation imbalance or ice

*S*

freezing,

*c*

*P*

is the specific heat of water,

coefficient of seawater at the surface,

/ *S*

1

/

*T*

is the thermal expansion salinity (Gill 1982), and *Q * is the net surface heat flux expressed as

*Q*

*Q*

*E*

*Q*

*L*

*Q*

*T*

*Q*

*B*

*Q*

*S*

The net surface heat flux affects the near surface turbulence by inducing a buoyancy flux. The distance from the surface where the wind stress and buoyancy are equally effective at producing turbulence is scaled by the Monin-Obukhov length (Eq. 22), and in the absence of salt fluxes, can be rewritten as (Lombardo and Gregg 1989).

*L*

*MO*

*u*

*

3

*J b*

0

*c p*

*u*

*

3

*gQ*

(28)

The similarity scaling works well when the near surface boundary layer is controlled vertically by horizontally uniform fluxes at the surface.

*L*

is used to separate the two

*MO*

13

asymptotic regimes: when turbulence, and when *z ***/**

*L*

*MO*

*z ***/**

*L*

*MO*

1 the wind stress dominates the production of

1 , the buoyancy controls the production.

**2.3.6. **

*Gas Fluxes *

The flow of inert gases between the atmosphere and ocean is controlled on a planetary scale by heat fluxes across the air-sea interface. Since the solubility of gases in water is temperature dependent, with cold water being able to dissolve more gas, areas of net heat input to the ocean degas to the atmosphere, while areas of net ocean cooling absorb the atmospheric gases (Schudlich and Emerson 1996). The gas flux across the air-sea interface is typically expressed as the product of a gas transfer velocity,

*k*

, and the difference between the gas concentration in air and water. In low and moderate wind speed, micro-scale wave breaking (or micro-breaking

1

) plays more significant role in the air-sea exchange. This microscale wave breaking on gas transfer can be modeled as

*k*

*u*

*

*S c*

1 2 where

is the fractional surface area covered by intense wave-induced surface divergence generated during the breaking process, velocity, and

*S c*

*D*

is a dimensionless constant,

*u*

*

is the friction

is the Schmidt number (the ratio of the kinematic viscosity of water to the gas diffusivity in water). For detailed information about the effect of microscale breaking wave on the air-water gas exchange the reader is referred to Zappa et al. (2001). Larger breaking waves cause whitecaps that inject air bubbles into the water and inversely, water drops and sprays into the air.

**3. Ocean Turbulence Measurement **

**3.1. Measurement systems **

Ocean near-surface boundary layer dynamics is dominated by high-frequency, intermittent phenomena that can only be described by turbulence statistics. Direct measurement of turbulent microstructure

2

in the ocean has proven elusive. Precise estimation of vertical fluxes of mass, heat, and momentum requires eddy correlation sensor packages comprising a current meter measuring the three components of small-scale velocity, and temperature and conductivity sensors sampling the same measurement volume. Indirect measurements, typically sampling the dissipation range of turbulence, are more typical. Some examples of indirect measurements and their applications are tethered or loosely tethered vertical microstructure profilers equipped with shear probes, thermistors, microstructure sensor packages installed on towed bodies etc.. In the following we give a categorized summary of turbulent measurement approaches. A detailed review on ocean microstructure measurements is given by Luek et al. (2002) and Thorpe (2007).

None of the measurement systems exemplified below is able to cover the full range of spatial scales from the smallest dissipative eddies (typically on the order of a centimetre) to the largest energy-containing eddies (typically on the order of several meters). Thus, to

1

Breaking of steep, wind-forced waves without air entrainment, which occurs at wind speeds well below the

level at which whitecaps appear.

2

The term microstructure is used for fluctuations associated with small-scale turbulence. The term fine structure is used for inhomogeneities related to stratification.

14

quantify statistics of turbulence, additional assumptions have to be made. The extrapolation to the four-dimensional space usually assumes local isotropy (that is, the turbulence has no preferred direction), at least for the smallest scales. To overcome the problem of sampling only a part of the turbulence spectrum, it is assumed that larger scales do not significantly contribute to the microstructure turbulence (for observations of the dissipation of TKE into heat) and that the smaller scales do not contribute to the energy-containing scales (for observations of Reynolds stress and kinetic energy). Taylor’s frozen turbulence hypothesis is often invoked, using a mean translation speed (e.g., an instrument’s profiling speed or the mean flow) to transform time series of turbulent fluctuations into spatial profiles. These assumptions are all only partly justified. Turbulence observations never have the accuracy achievable for mean properties such as temperature or currents (Burchard et al. 2008b).

The technology has matured considerably during the past 40 years of oceanic turbulence profiling. Many profilers have been developed and their use has revealed numerous unexpected features of oceanic processes. However, a complete review of all these instruments would be impossible from the scope and the frame of this report. However, we provided ample references in this section to give the reader the opportunity to examine the many details for the sake of brevity.

*Point measurement at spatially fixed positions*

: The most widely used method to sample velocity fluctuations in the water column uses the acoustic Doppler principle, either at single point using an acoustic Doppler velocimeter (ADV) or over a limited vertical range using a high-frequency acoustic Doppler current profiler (ADCP). Such instrument is an active sonar that transmits acoustic signals (by one transmitter) and then receives their echoes (by three or more receivers) partially reflected by sound scatterers in the water, like plankton or other floating particles. The scatterers are usually passively advected by water motion. From the measured Doppler shift between emissions and echoes, the flow velocity is deduced.

An example ADV is shown in figure 3-a. An important advantage of an ADV is that it measures the flow in a small sampling volume. This enables measurement to be taken without interfering with the flow (Nikora and Goring 1998). Advantage of an ADCP is that it provides the 3D velocity profile over a limited depth range, in contrast to an ADV’s point measurement

(Gargett 1997; Souza 2007). Three transducers (beams) slanted at the same angle from the horizontal are sufficient to infer the three orthogonal components of the velocity.

An additional four beam, when available, allows velocity to be inferred when one beam fails and also provides for an error velocity estimate. Recent ADCPs with five beams are particularly suited for Reynolds stress measurements since the vertical velocity can be inferred with high accuracy (for an application see Peters et al. 2007). Figure 4-a shows a conventional ADCP with four beams. A detailed outline of the procedure to calculate a three dimensional current vector based on the measured velocities of a three-beam ADCP is illustrated in figure 4-b. Reynolds stresses are measured in this instrument by the use of statistical properties of the measured in-beam velocity fluctuations which is known as the variance method (Lu and Lueck 1999).

15

**Figure 3.** a) Schmatic of an ADV instrument (NortekUSA 2010). b) Sketch of TAMI and the mooring arrangement used in Cordova Channel (Lueck and Huang 1998).

**Figure 4.** a) Schmatic of typical 4 beam ADCP sensor head. The red circles denote the 4 transducer faces ( http://oceanexplorer.noaa.gov/technology/tools/acoust_doppler/media/adcp.html

). b) Beam geometry and calculation of the horizontal and vertical velocity components from the measured inbeam

**(**

*b*

1

**,**

*b*

2

**,**

*b*

3

**)**

for a three-beam ADCP (Lorke and Wuest 2005).

16

The Tethered Autonomous Moored Instrument (TAMI, Fig. 3b) is an example of moored turbulence measurements at a fixed point by use of airfoil shear probes (Lueck et al. 1997).

The instrument measures the microstructure shear in the dissipation range of the wavenumber spectrum using shear probes, and the temperature fluctuations using fast-response thermistors.

It is a convenient platform to sample the background temperature and salinity, horizontal currents, in this example using two ducted rotors and a fluxgate compass. The body motions are monitored using accelerometers and a pressure transducer. With today’s technology and data storage capacity, several months of raw data sampling is possible. TAMI, however, burst-sampled for 128 second every 5 min and stored reduced data including and bandaveraged spectra and statistical parameters (Lueck et al. 1997; Lueck and Huang 1998).

*Towed and free rising or falling microstructure profilers*

: The microstructure profilers are usually equipped with sensors to measure the fluctuation of temperature, conductivity, and velocity (current shear) in the microstructure scale range. The profilers are typically also equipped with accurate CTD (Conductivity, Temperature, Depth) sensors. Some specialized profilers are equipped with sensors for background horizontal current, oxygen concentration or fluorescence. On the *Chameleon* microstructure profiler a pitot tube was installed to measure vertical velocity which allowed for eddy correlation measurements from a vertical profiling instrument (Moum 1996). In their thorough review, (Lueck et al. 2002) give a historical summary of all microstructure profilers (both towed and vertical profiling).

Dissipation rate profiles resulting from two sets of different profilers with different electronics and data processing schemes have been contrasted in Moum et al. (1995). for two American profilers and in Burchard et al. (2002) for two European profilers. In addition to instrumentation developed in various pioneering research institutions, a couple of microstructure profilers are commercially available: the VMP series by Rockland Scientific

International, Canada ( http://www.rocklandscientific.com/ ) and the MSS series by Sea and

Sun Technology and ISW Wassermesstechnik, Germany ( http://isw-wasser.com/prandke/ ).

The latter also provides profilers for rising type measurements. For the MSS series, adjusting weights can be fixed at both ends of the profiler housing to alter the buoyancy of the profiler.

For rising measurements, the microstructure profiler is given positive buoyancy. For vertical sinking measurements, the profiler is balanced with slightly negative buoyancy. For all loosely-tethered microstructure profiler systems, effects caused by cable tension (vibrations) and the ship's movement are minimized by paying out enough cable to keep it slack.

Rising profilers are needed to study the turbulence in the near-surface boundary layer. The uppermost meters are then satisfactorily resolved. A particularly relevant example is given in

Ward et al. (2004). Another example is the near surface turbulence instrumentation employed in the western equatorial Pacific during the TOGA Coupled Ocean–atmosphere Response

Experiment (COARE). The system consists of a free-rising profiler, bow-mounted sensors, and a dropsonde (Soloviev et al. 1999). An advantage of this method is the absence of rigid mechanical connection with the ship’s body and is an effective method for measuring the near surface turbulence. Figure 5 schematically shows deployment of the devices and includes photographs of the free-rising profiler and the bow sensors. The free-rising profiler fitted with microstructure shear, temperature, and conductivity probes is connected to a shuttle (Fig. 5c,d) which allows the device to slide from the ship into the water. The profiler sinks a distance about 15 to 35 m far from ship’s wake. At a depth of about 20 m, the pressure-release mechanism releases the profiler from the shuttle and the profiler starts ascending at a speed

17

depending on the net buoyancy of the profiler. Due to presence of surface waves and pitching of the vessel, the bow probes are used to scan near surface boundary layer. The bow probes include high resolution probes of pressure, temperature, conductivity, and acceleration sensors which resolve the near surface turbulence as the ship steams (Fig. 5-d) (Soloviev and Lukas

2006).

**Figure 5.** Schematic diagram showing the near surface turbulence measurement employed by Soloviev and

Lukas. (2003a): a) deployment and d) the probe mounted on the bow. Here, 1- free-rising profiler coupled with shuttle; 2- temperature, conductivity, and shear probes; 3- shuttle; 4- bow frame; 5- bow units (temperature, conductivity, and pressure sensor, shear probe, and tilt sensor); 6- dropsonde; 7- temperature probe of microwire type. The photographs show b) the bow sensors, c) the free-rising profiler with the shuttle, and e) the sesnors on the profiler. (Soloviev and Lukas 2006).

18

*Particle image velocimetry (PIV) in a two-dimensional plane*

: Turbulence energy is estimated in a backward facing step flow with three-component PIV. Estimates of turbulence energy transport equation for convection, turbulence transport, turbulence production, viscous diffusion, and viscous dissipation in addition to Reynolds stresses are computed directly from

PIV data (Piirto et al. 2003). Figure 6 shows a schematic of submerged components of an oceanic PIV system. The light is delivered by a dual-head, pulsed dye laser through an optical fiber to a submerged optical probe, where the beam is expanded to a sheet that illuminates the sample area. The submersible system also contains a Seabird electronics, optical transmission and dissolved oxygen content sensors, a pressure transducer, a biaxial clinometers, and a compass. The platform is mounted on a hydraulic scissor-jack to enable acquisition of data at various elevations above the seafloor. The platform can also be rotated to align the sample area with the mean flow direction (Doron et al. 2001).

**Figure 6.** Schematic of oceanic PIV system: a) surface mounted laser, control and image acquisition subsystems, b) side and top views of the submersible instrument (Doron et al. 2001).

*Example field experiments:*

The field experiments designed to sample near surface turbulence and air-sea interaction show evidence of coupling between the surface temperature, the surface turbulent velocity fields, and the surface waves. Figure 7 shows an example of a field experiment set-up conducted from research platform (R/P) *Floating *

*Instrument Platform *

(*FLIP*), moored approximately 150 miles off the coast of southern

19

California (Veron et al. 2009). The main instruments are composed of an integrated active and passive infrared imaging, a laser altimetry system, a 6-degree-of-freedom motion package, and an eddy covariance system to acquire meteorological and atmospheric boundary layer flux data (not covered in this report).

In the presence of strong wind in the open ocean, breaking waves are an active process for generating of turbulence. For observing the turbulent velocity and bubble field as a part of air-sea fluxes, Gemmrich and Farmer (2004) conducted the Fluxes Air-Sea Interaction and

Remote Sensing (FAIRS) experiment aboard the R/P *FLIP* 150 km offshore of Monterey,

California. A surface float, tethered to the starboard boom, supported three orthogonal 2-MHz pulse-to-pulse coherent acoustic Doppler sonars and two acoustical resonators as well as an environmental package, monitoring water temperature, salinity, and float tilt and heading

(Fig. 8). Three components of velocity profile are determined by Dopbeams which contains three passive sonars.

A set of eight 100-kHz side-scan sonars, mounted on *FLIP*’s hull at depth ranging from 15 to 91.5 m, is used to estimate the directional wave field as well as the surface drift speed (Fig. 8-a).Video recording also is used to verify the occurrence and estimate the size of breaking waves at the float. The free-flooding resonators are used to determine the size distribution of microbubbles generated by breaking waves (Fig 8-b).

**Figure 7.** Instrumentation setup from R/P FLIP (Veron, 2009).

**Figure 8.** a) Schematic of the surface-following float contains two resonators and the head of the vertical

Dopbeam. b) Sketch of the deployment setup. The float was tethered to the starboard boom of R/P *FLIP*.

(Gemmrich and Farmer 2004).

20

**3.2. Sensor Features **

Technological advances in the last decades, including novel instrumentation, greater capacity of data recorders, faster recording rates, and more reliable and smaller sensors have increased our measurement capability, and understanding of turbulence in the ocean. Airfoil shear probes to measure velocity shear, fast-tip thermistors to measure temperature at subcentimeter scales, and free falling (rising) instruments to carry sensors are examples of such advances.

**3.2.1. **

*Temperature *

Small scale temperature fluctuations were the first to be measured to gather insight into mixing process in the near surface layer (Gregg et al. 1973; Osborn and Bilodeau 1979).

Today various sensors are commercially available. The SBE 8 microstructure temperature sensor is a lightweight fast-response sensor intended for use in marine profiling applications where its high speed and spatial resolving power offer the ability to characterize small scale ocean temperature features (Fig. 9).

The sensing element is a remote-cabled, probe-mounted thermistor (thermometrics type FP07). The thermistor voltage output is pre-emphasized, so the sensor's output increases as a function of the frequency components in the temperature signal. The effect of pre-emphasis is to magnify the sensor output for rapidly changing temperature, thereby overcoming the restrictions on system resolving power that would otherwise be imparted by the use of conventional (e.g., 16-bit) digitizers. The SBE 8's preemphasis response magnifies a 20 Hz temperature signal by a factor of 200, facilitating acquisition of signals 200 times smaller than could be characterized by conventional CTD sensors (Kanari 1991; Gregg 1999).

.

**Figure 9. **The SBE 8 integrated instrument probe, cabling and electronic housing.

(http://www.seabird.com/products/spec_sheets/8data.htm)

**3.2.2. **

*Conductivity *

The microstructure conductivity sensor has been developed for making very rapid, high resolution measurement of electrical conductivity of water. Several versions exist, for example with dual needle or 4-electrodes. The latter contains four-electrode (platinum spheres) conductivity sensors supported by fused glass adopted for measurement of the mean conductivity profile and high frequency conductivity gradient profile (Fig. 10). The electronic circuit makes 4 terminals measurement of the conductance of the water by supplying an AC

21

**Figure 10. **Diagram of microstructure conductivity sensor

(Nash and Moum 1999)

. current between the inner electrodes of the sensor and measuring the AC voltage that develops across the outer electrodes. The ratio of this current to voltage is computed by circuit and a representative voltage supplied at the circuit output. When used in this way, the output voltage corresponds to a weighted volume average of the conductivity of the water in the vicinity of the sensor electrodes (Nash and Moum 1999).

**3.2.3. **

*Shear *

An airfoil shear probe is a piezoceramic beam that generates small voltages as the turbulent velocity varies the lift and thus the bending force on an airfoil as it moves through the water (Fig. 11). It can measure velocity fluctuations on a spatial scale of about 10 mm.

The general behavior and the construction principles of an airfoil sensor have been described in detail by Osborn (1974).

The basic scheme of the shear sensor operation is shown in figure 10. The cross force per unit length due to potential flow for an axially symmetric airfoil in a water jet of speed

*U*

(with horizontal,

*u*

, and vertical,

*w*

, velocity components) and an angle of attack

can be expressed as

** Figure 11. **The piezoceramic airfoil shear probe operation in presence of cross force (Lueck 2005b).

22

*f p*

1

2

*U*

2

*dA*

sin

*dx*

2

(29) where

is the density of the fluid and / is the rate of change in airfoil cross section area in the axial direction. By integrating from the tip of the sensor to the position where

/

0 , and by elimination of

from equation (29), the total cross force (lift force) is obtained as

*F*

*A w u*

where

*u*

*U*

sin

and

*w*

*U*

cos

are the horizontal and vertical components of the flow.

However, the speed of the probe in water also contributes to the cross-force. The constant axial velocity component

*w*

is due to falling or rising speed of profiler. The lift force, sensed by the strain-gauge, is converted to an electric voltage force. Time derivative of output signal

*E s*

*dE a dt*

*S w*

2

*du dz*

where

*S*

is the overall sensivity of the linear response of the probe and *E * is the output

*s*

voltage. Consequently, the vertical shear of horizontal flow has the following form

*du*

*dz*

1

*S w*

2

*E s*

There are several sources of errors in this type of shear measurements, including limitations in the mathematical assumptions, determination of profiling velocity, errors in the calibration procedure, drift of the shear-probe sensitivity, temperature and pressure effects and interference from profiler vibrations (Burchard et al. 2008a). Furthermore, the wavelengths smaller than the airfoil’s spatial response** **length cannot be resolved (Ninnis 1984; Prandke and Stips 1998; Lueck 2005a). The shear is sensitive to the sink/rise velocity, which poses a challenge in measuring in the near surface layer influenced by wave orbital velocities.

**3.3. Parameter Estimation **

**3.3.1. **

*Turbulence Dissipation Rate *

In the TKE balance equation (1), the major sink term is the dissipation rate, defined as

2

*ij s ij*

(30) where, the second order tensor deformation rate of fluid particles due to the velocity fluctuation gradients field, and

is the kinematic viscosity

3

. For isotropic turbulence equation (30) can be expressed as (Osborn

1974)

3

Viscosity is weekly dependent on salinity and pressure but strongly dependent on temperature. For estimating it the following polynomial function is widely used

**(**

1

**.**

79247

0

**.**

05126103

*T*

0

**.**

0005918646

*T*

2

**)**

10

6

.

The typical value for water at 20 o

C is about 10

-6

m

2 s

-1

.

23

7

**.**

5

*d u dz*

2

(31)

Isotropy can be assumed without any restriction in homogeneous flow, except close to boundaries. But in a stratified flow, like that in the thermocline, the stratification modifies the dynamics of turbulence (Rehmann and Hwang 2005). The effects of stratification can be inferred from the buoyancy Reynolds number (Eq.11),

*N*

2

(Yamazaki and Osborn 1993).

Anisotropy becomes more pronounced as

diminishes relative to the stratification. The isotropic formula (Eq. 31) can be used with an error of about 35% for

*N*

2

20 , and isotropy is achieved for

*N*

2

200 (Yamazaki and Osborn 1993).

**3.3.2. **

*Vertical Eddy Diffusivity for Mass *

Vertical eddy diffusivity for mass, density gradient ,

*d*

*dz*

,

*w*

*K*

*d*

(32)

*dz*

where

and

*w* are the fluctuations of density and vertical velocity, and an overbar denotes averaging. When turbulence is steady and homogenous, a balance is obtained between the turbulent shear production, losses to viscous dissipation and the buoyancy flux *B * as

*u w*

*u*

*z*

(33)

By the use of equations (31), (32) and (9), equation (33) yields

*K*

*N*

2

(34) where

is mixing efficiency factor defined by equation (10). This approach is called the

Osborn model (Osborn 1980).

**3.3.3. **

*Dissipation Rate of Temperature Variance and Eddy Diffusivity for Heat *

For steady, homogenous, and isotropic turbulence, temperature variance equation (3) can be simplified to

2

*T*

*w*

*T*

*z*

*T*

(35)

Kinematic turbulent heat flux can be written as the product of the eddy diffusivity for heat,

*K*

*T*

, and the mean temperature gradient

*w*

*T*

*K*

*T*

*T*

*z*

From this and using equation (14), the eddy diffusivity for heat is obtained as

*K*

*T*

3

*k*

*T*

*T*

*T*

*z*

*z*

2

2

24

where the factor 3 is valid for complete isotropy. In the absence of direct measurement of microscale velocity shear, the assumption

*K*

*K*

*T*

is often made to determine

and buoyancy flux. This method is called the Osborn-Cox model (Osborn and Cox 1972; Ardhuin et al. 2009).

**3.3.4. **

*Vertical Overturns *

In stratified flow, turbulent mixing has manifestations on the density profile that can be used to infer the turbulent lengths scale and dissipation rate. Shear instabilities and Kelvin-

Helmholtz billows lead to gravitationally unstable overturns. The characteristic of these instabilities can be obtained from the density profile relative to a reference density profile

*m*

computed by sorting the original density profile

(Fig. 12). From these density profiles, the density fluctuation, displacement *d *

*m*

and the vertical displacements (Thorpe

) can be computed. Thorpe displacement is the minimum distance a fluid parcel needs to be moved from the observed profile to produce the synthetic stable profile

(Thorpe 1977; Piera et al. 2002).

The Thorpe scale can be calculated as the root mean square of the Thorpe vertical displacements as

*L*

*T*

*d *

2

.

The Ozmidov scale (Eq. 6) is the scale at which buoyancy effects are felt strongly by the turbulent eddies. It represents the vertical scale at which buoyancy and inertial forces are equal. The Thorpe scale is found to be proportional to the Ozmidov scale (

*L*

*O*

*L*

*T*

0 .

8 ), which allows for an estimate of

through conventional CTD measurements of sufficient vertical resolution (Dillon 1982).

**Figure 12. **Schematic of different steps for deriving vertical displacement from reference potential density,

*m*

, and original potential density,

(Fernandez 2001).

25

**3.4. Measurement Limitations and Platform Motion **

In recent years a great deal of attention has been directed toward making high-resolution measurements of turbulence statistics at sea. The problems largely arise from three sources: undersampling, platform motion, flow distortion, and environmental factors unique to the ocean (Edson et al. 1998).

Vertical microstructure profiling has demonstrated the intermittency of ocean mixing in time, depth and intensity. Due to the logistics of vertical profiler deployment, these data sets provide sparse horizontal sampling. In spite of this intermittency, by employing sufficient sampling and averaging, relations can be found between external forcing and dissipation in the mixed layer. For example, the integral of dissipation over the mixed layer was found to account for about 2% of the energy flux from the atmospheric boundary layer (Ardhuin et al.

2009). In addition, the intensity of the dissipation was found to be well correlated with the cube of the wind speed (Greenan and Oakey 1997). In the surface layer, wave orbital velocities can dominate the turbulent fluctuations. The velocity scale of turbulent fluctuations in the near-surface layer of the ocean is about 1 cm s

-1

, while the typical surface wave orbital velocities are 1m s

-1

. The energy of the disturbance due to surface wave orbital velocities is four orders of magnitude higher than those of the turbulence signals. The additional complication is that the time scales of the surface waves and the near-surface boundary layer turbulence substantially overlap. The linear statistical filtering that has been widely used to separate linear waves and turbulence from moored or tower-based velocity measurements cannot remove the non-linear components of surface waves, which may result in an overestimation of the turbulence dissipation rate (Soloviev and Lukas 2003a). Another traditional limitation of sinking vertical profilers is the inability to sample the upper 5-8 m of the ocean mixed layer due to the fact that the profiler typically starts about 2 meter below the ocean surface and requires a few meters of free-fall before it stabilizes. Furthermore, the instrument descent starts close to the ship so the measurements near the surface are contaminated by the effect of the ship on the surrounding environment (Greenan and Oakey

1997).

The platform motion contaminates the velocity fluctuations and imposes problems when estimating the turbulence fluxes. This motion contamination must therefore be removed before computing of turbulence fluxes. The contamination arises from three sources: 1- the pitch, roll, and heading variations of the platform during measurement, 2- angular velocities at the microstructure due to rotation of the platform about its local coordinate system axes, and

3- translational velocities of the platform with respect to a fixed frame of reference. In the next section, a brief discussion about rotational frame system with 6 degrees of freedom and multivariate correction procedure with 3 degrees of freedom is given to remove vehicle motion and vibration contamination.

**3.4.1. **

*Platform Motion and Vibration Correction Procedures *

A variety of approaches has been used to correct for the platform motion contamination on velocity measurements (Edson et al. 1998). One approach is to correct platform motion by carefully measuring this motion relative to a reference frame. Therefore, turbulence measuring instrumentations must be equipped with extra sensors such as accelerometers and gyros. Signals obtained from both types of instrumentations are the processed to produce corrected quantities of interest at the measurement location.

26

Assume that the turbulent momentum fluxes are measured by the eddy correlation method. The platform motion contaminates the velocity fluctuations and this motion contamination must be removed before computing the fluxes (Levine and Lueck 1999). The true velocity vector can be written as

*V true*

**T**

1

*V obs*

**T**

2

*V equip*

*V trans*

(36) where,

*V true*

is the desired velocity vector in the reference coordinate system,

*V obs*

is the measured velocity in the platform frame of reference,

*V*

is velocity of measurement

*equip*

instrumentation in the platform frame.

*V*

is the translational velocity vector at the center of

*trans*

motion of platform to a fixed coordinate system, and

**T**

1

and

**T**

2

are rotation transformations of velocity vector and instrument velocity from the platform frame to the reference coordinate system (Miller et al. 2008). When the motion sensors and platform have the same coordinate system,

**T**

**1**

**T**

**2**

.

*V equip*

has a linear and an angular component as where and

*a*

*a x*

*y*

**,**

*z*

**Ω**

*obs*

*x*

**,**

*y*

**,**

*z*

*V equip*

*a*

*dt*

**Ω**

*obs*

r (37)

is the linear acceleration (accelerometer time series output)

is the angular velocity vector of platform coordinate system that is given as

**Ω**

*obs*

*obs obs*

*obs*

where, the subscript “obs” denotes measurements made in the platform frame of reference, the over-dote denotes time derivative, and

*obs*

**,**

*obs*

and

are roll, pitch and yaw, respectively

*obs*

(Euler angles, Fig. 13). The yaw is defined positive for left-handed rotation. The negative sign

**Figure 13. **Location of the platform (body) and fixed coordinate frames (Linklater 2005).

27

in front of the yaw rate is required to compute the angular velocities in the standard righthanded coordinate system. The vector r

r

1

r

2

is the position vector of the velocity sensors with respect to motion package. The vector r

1

is the position vector of velocity sensors with respect to the centre of gravity, and r

2

is the correction term when motion measurement system is not located at the centre of motion.

The rotation of coordinate system *xyz * into reference frame

*x*

*y*

*z*

is stated by transformation matrix

**T**

**T**

1

**T**

2

as

*k i j*

**T**

*i k j*

where

**T**

*i*

*j*

**(**

*i*

**.**

*j*

**)**

is direction cosine. The rotation matrix has six independent relations among the nine elements. Some properties of the matrix are that its inverse is its transpose, and its determinant is unity. Meanwhile, successive rotations are not commutative (Thwaites 1995).

If

*A*

and

*A*

are the roll and pitch matrices, thus

*A*

**(**

**)**

*A*

**(**

**)**

*A*

**(**

**)**

*A*

**(**

**)**

If rotations with respect to roll, pitch and yaw are labeled by 1, 2 and 3 respectively, there are

12 possible sets identified by the axes of rotation such as 123, 321 and etc. (Pio 1966).

Therefore, the rotation transformation, **T **, has several forms. For example, in the 321 class of

Euler angles, the rotation matrix **T ** is given as

*A*

**T**

cos sin

0

sin

cos

0

0

0

1

**,**

*A*

cos

0 sin

cos

cos

*A*

sin cos

sin

sin

sin

cos

0

1

0 cos

sin

sin

sin

sin

0 cos

**,**

cos

*A*

1

0

0 cos

sin

sin

sin

sin

0 cos

sin

sin

sin

sin

0 sin

cos

**,**

cos

cos

cos

0 sin

cos

(38)

In the fixed frame, only yaw is measured, pitch is measured in a yaw frame, and roll is measured in a pitched and yaw frame. From equation (38), Euler angle rates can be derived as

**Ω**

*true*

**TΩ**

*obs*

0

0

*obs*

*A*

**(**

**)**

0

*obs*

0

*A*

**(**

**)**

*A*

**(**

**)**

*obs*

0

0

*J*

*obs obs*

*obs*

(39) where

*J*

0

0

1 sin

*t*

an

cos sin

cos

cos

sin cos

cos

tan

For small angles, equation (39) can be approximated as

28

*obs obs*

*obs*

*obs*

*obs*

*obs*

(40)

The above mentioned technique is based on a rotation matrix with 6 degrees of freedom. Another technique is based on removing platform motion effects and vibrations using the standard spectral signal processing techniques (Goodman et al. 2006). In this approach, the acceleration measurements are used to minimize the contamination of the shear and temperature probe measurements by vehicle motions and vibrations of the probe mounts.

Let the matrix

**s**

*obs*

*u*

**,**

*v*

*T*

represent the time series of the rate of change of horizontal velocities, and temperature measured by the shear probe, and thermistor. Furthermore, let

*a ***,**

1

*a*

2

and *a *are time series of accelerometer output with respect to

3

*x***, ** and *z * directions, respectively. True signal can be represented as

**s**

*i*

**,**

*obs*

**s**

*i*

**,**

*true*

*b ij*

*a j*

(41) where,

*b * is the impulse response, the operator

denotes convolution, matrix

**s**

*true*

(

*i*

**s**

is uncontaminated signal related to i th

*i***,**

*true*

component of uncontaminated

**,**

2 3 ). Further, assume in the equation (40), platform motions and vibrations are statistically independent that is

**s**

*i*

**,**

*true a j*

0

**,**

for all

*i*

**,**

*j*

(42)

For motions with scales comparable to and longer than the length of vehicle, this assumption breaks down because instrumentation response to such motion. In the Fourier transform domain, equation (40) is written as

**S**

*i*

**,**

*obs*

**S**

*i*

**,**

*true*

*B ij*

*A j*

(43) where the capital cases represent the Fourier transform of their lower case counterparts, and

*B ij*

*B ij*

is the frequency transfer function relating the probe signals to the accelerometer signals. By multiplying equation (43) by its complex conjugate, ensemble average

(

**S**

*i*

**,**

*obs*

**S**

*****

*j*

**,**

*obs*

), and use of equation (41), it then follows that

**φˆ**

*ij*

**φ**

*ij*

*ik*

*kl*

1

*lj*

*****

(44) where,

**S**

*i ***,**

*true*

and

**φˆ**

*ij*

is the cross-spectrum of the contaminated signal

**S**

, the

*i***,**

*obs*

**S**

*i*

**,**

*true*

**S**

*****

*j*

**,**

*true*

, the

**φ**

*ij*

**S**

*i*

**,**

*obs*

**S**

*****

*j*

**,**

*obs*

is the cross-spectrum between

*ij*

the *A * and observed signals, the

*j*

*B ij*

*il*

*lj*

1

**,**

*A *, and

*j*

*ij*

*****

*B*

*****

*ji*

*jl*

1

*li*

*****

(45)

Equation (44) is used to correct both the spectra series are corrected by convolving the accelerometer signals with the weighted function *b *

*ij*

which is obtained from inverse Fourier transform of equation (45).

29

**3.4.2. **

*Despiking and Denoising *

Spikes resulting from hits of plankton or other solid matter are inherent in every microstructure profile. Therefore, the first and most important step during the data processing, of e.g. shear probe data, is the spike removal. A typical procedure applied with profiler data is an iterative calculation of the signal variance over a certain record lengths and the removal of data that are larger than a specified value times the standard deviation.

Microstructure data are contaminated by all kinds of mentioned noise resources. Thus, cleaning the data is an important step that can be done by several techniques (Rodriguez

2005) such as

classical digital filters: Both low frequency motions of profiler and thermal drift of airfoil probe lead to low frequency noise of microstructure profiler. For cleaning these component of noises, classical filters such as Butterworth filter are used. Furthermore, high frequency electronic noises is removed by digital filters. Mechanical resources of noise can be removed by band path filters such as Lanczos filter.

wavelet denoising: Wavelets have an important application in signal denoising. After wavelet decomposition, the high frequency subbands contain most of the noise information and little signal information. In this level, soft thresholding is applied to the different subbands. The threshold is set to higher values for high frequency subbands and lower values for low frequency subbands.

**4. Turbulence Numerical Modeling **

There are various strategies for modeling turbulence, each of them resolving a different interval of the spectrum of the flow dynamics (Davies et al. 1995; Burchard et al.

2008a). In this section, a brief overview based upon simple eddy viscosity closure or the use of turbulence energy models is presented.

**4.1. Direct Numerical Simulation **

The large size of the ocean circulation requires decomposition of ocean flows into various scales. In direct numerical simulation (DNS), the Navier–Stokes equations are directly solved by means of discretization techniques for all scales of motion of a turbulent flow. The

DNS requires very high spatial and temporary resolution that it is beyond simulation by the computer available at present time.** **

**4.2. Large Eddy Simulation **

In LES, the largest eddies are computed, and the effect of the smaller eddies on the flow is represented by using sub-grid scale (SGS) models. The main underlying concept is the striking feature of a turbulent flow field that, while large eddies migrate across the flow, they carry smaller-scale disturbances with them. Large eddies carry most of the Reynolds stress, and can be different in various types of flows, and thus must be computed. Smaller eddies, on the other hand, contribute much less to the Reynolds stress, and may possess a relatively more universal character. The fundamental idea is to compute the large eddies, which carry out most of the mixing and to model the smaller eddies, which dissipate the energy cascading from the larger scales. In this way, the resolution of the smallest dissipative scales is avoided, and this can potentially reduce the computational cost and/or be used to increase the Reynolds

30

number of the simulations (Ozgokmen et al. 2007). In oceanography, this method is usually applied for investigating near-surface mixed layers, mostly with horizontally homogeneous macroscopic conditions. Wave-driven Langmuir circulation, buoyancy-driven thermal convection and shear-driven Kelvin-Helmholtz billows are three dominant large eddies in the ocean surface mixed layer that can be solved successfully by LES (Li et al. 2005).** **

**4.3. Statistical Turbulence Modeling **

Reynolds averaging of the Navier Stokes equations lead to the unknown Reynolds stresses with different orders. For representation of vertical turbulence, these unknown stresses can be parameterized by means of turbulence closure techniques. These closure techniques can be divided into three major model classes (Burchard and Baumert 1995). 1- bulk models which use strict homogeneity of mixed layer; 2- one-equation closure which resolve TKE in the vertical, based on an algebraic or integral formulation for eddy length scale; and 3- two-equation closures which use an additional vertical equation for the length scale or a related quantity. The most popular closures of this group are Mellor-Yamada

(Mellor and Yamada 1982) level 2.5 model and

*k*

model.

**4.4. Empirical Turbulence Modeling **

The models of this group are originated from the Reynolds-averaged Navier–Stokes equations similar to statistical turbulence models. Instead of approximation of turbulence fluxes by employing closure techniques, they use entirely empirical knowledge of fluxes.

Thus, the accuracy of these techniques is directly related to accuracy of measurements

(Burchard et al. 2008a). The non-local K-Profile Parameterization (KPP) is a widely used vertical empirical turbulence model.

**5. Summary **

Physical ocean properties and variables near the sea surface, particularly dynamical quantities such as currents, turbulence, and turbulent fluxes, are influenced significantly by ocean surface waves. In addition, movements of instruments and measurement platforms, which will be unavoidable on floating platforms which move with the waves, often bias the measurements, even if the instantaneous motions of the platform are corrected for. This complex nature of the near-surface layer of the ocean is covered in different sections of this report. Examples of measurement techniques and sensors are given with references to more detailed reviews. Finally, a short description of different numerical models was presented for simulation of turbulence in the upper ocean boundary layer.

31

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