Schematic overview of boundary layers

SEMIN AR — ATMOSPHERE–OCEAN INTERA CTION The turb ulent boundary lay ers abo v e and belo w the air –sea interface

Introduction

• The flow of air and water, respectively above and below the water surface, is in general turbulent

– except for thin laminar layers immediately adjacent to the surface

• fluid flow in the near-surface boundary layers is often assumed to have properties similar to those of turbulent flow near a solid surface

– Scaling laws: Prandtl, von Kármán, Kolmogorov, Monin/Obukhov – Modifications: stratification, convection, Coriolis force

• Fluxes:

– Momentum, heat, mass

• Measurements

• Modelling

• Wrap-up

Schematic overview of boundary layers

Flow approx. geostrophic Ekman layer

"Constant stress" layer

"Mixed layer"

"Ekman layer"

Thermocline

Flow approx. geostrophic Laminar layers

’00 m

~ mm

’ − ’0 − ’00 m

’0 − ’00 m

’000 m

Water surface

Geostrophic flow

• Flow in the ‘interior’ of the atmosphere and the ocean

• Balance between dominant pressure gradient and Coriolis forces u_h = 1

f ˆz×∇_hp (1)

• Density gradients give ‘thermal wind equation’:

∂u_h

∂z = − g

fρˆz×∇_hρ (2)

• Additional accelerations: cyclostrophic flow, inertial oscillations, Rossby waves, etc.

• Turbulent motions have a minor effect on the dynamics (but may be significant for diffusion of heat and mass).

Ekman layers

• Balance between Coriolis force −f ˆz×u_h and (turbulent) friction ∂/∂z(K_v∂u_h/∂z), leading to a mean velocity vector which traces a helical path.

• A transitional zone between near-geostrophic flow and a more intensely turbulent flow nearer the water surface

• The vertical flux of momentum via turbulent friction, K_v∂u_h/∂z, is not independent of the vertical coordinate, even if the mean flow is time-independent.

• In the ocean in particular, the Ekman layer may be difficult to resolve from measurements, because of large flow variability from other causes.

• Nevertheless, the overall dynamical effect of the Ekman flow may be well defined: for example, the Ekman layer transports fluid at a fixed rate −ˆz×τ/(ρf), proportional to and at right angles to the interfacial stress τ:

‘Constant flux’ layer

• As we approach the surface from above, the turbulent forces become dominant, the mean vertical momentum flux becomes approximately constant, and the mean flow may be

considered to have a Prandtl—von Kármán wall layer behaviour:

u = u_∗

κ log z z₀

, (3)

where u_∗ = τ/ρ, κ ≈ 0.4 is the von Kármán constant, and the small quantity z₀ is the roughness length, being derived empirically, but for solid rough surfaces being

proportional to the size of the roughness elements.

• The roughness length over the sea is variable, and in general of the order of the roughness length of smooth land surfaces (e.g. grass). It appears to be dependent on the wind speed (or wind-wave slope), for example in the Charnock (1955) relation the roughness length depends on the friction velocity:

z₀ = αu_∗²/g, (4)

with α having a value of between 0.01 and 0.02.

‘Constant flux’ layer in the ocean?

• If there is a constant flux turbulent boundary layer in the ocean which transfers the turbulent stress applied by the atmosphere, we will have

τ = ρ_au_∗a² = ρ_wu_∗w². (5)

• Hence u_∗w/u_∗a = (ρw/ρa)^−1/2.

• This leaves the question of the roughness length in the water open. Alternative approaches appear to be:

– Employ turbulence closure schemes, involving differential equations for a

combination of turbulent kinetic energy, mixing length, turbulent energy dissipation, etc.

– Use a ‘slip layer’ formulation (‘Inertial coupling’, J. A. T. Bye 1986ff) – Use a ‘Charnock’ formula with a large coefficient, e.g. O(10²)

– Introduce an ‘arbitrary’ roughness length to fit measurements/what you want to see – . . .

Ocean mixed layer

• The part of the ocean which is disturbed by what goes on at the surface

• Often needs to be treated differently in numerical models than the deeper ocean

• Can for some purposes be assumed to have a ‘slab’ behaviour, with uniform

temperature/salinity/(potential) density etc., with inputs/outputs by bulk formulae at the surface, and mixing using an entrainment coefficient at the bottom boundary

(thermocline). Models by R. Weller, R. Pinkel, etc.

• However, the current may not be vertically uniform, even if other variables are (due to wave-induced Stokes drift, turbulent ‘wall layer’ behaviour, and so on).

• Treatment may be complicated if there is more than one thermocline (e.g., diurnal/summer heating in the upper 1–2 m)

Stratification, convection

• Vertical fluxes may be enhanced/suppressed where the vertical gradient of the density is non-zero

• Or, alternatively, the vertical buoyancy (= −density) flux influences both the turbulence intensity and the density gradient

Stratification, convection

• Relevant quantities:

– Gradient Richardson number

Ri = (g/ρ)∂ρ/∂z

|∂u/∂z|² (6)

– Flux Richardson number, given by the ratio of the buoyancy flux to the rate of production of turbulent kinetic energy by shear:

Rf = (g/ρ)ρ⁰w⁰

u_h⁰w⁰∂u_h/∂z (7)

– Monin-Obukhov length h_mo, various formulae including that of Stigebrandt (1985 JPO):

h_mo ≈ 0.3u_∗³/B, B = buoyancy flux (8) (the density stratification may be neglected at vertical scales much smaller than h_mo)

• The logarithmic behaviour of the mean velocity profile in a ‘wall layer’ may be amended by including an empirical function of z/h_mo: this empirical function has been determined by a series of experiments in the atmosphere over land (mostly in Kansas)

(From Nerheim & Stigebrandt JPO 36:1591–1604, 2006)

(From Nerheim & Stigebrandt JPO 36:1591–1604, 2006)

Laminar layers, heat and mass flux

• The coefficient of air–water transfer of momentum is considerably greater than the

corresponding transfer coefficients for heat and mass, as momentum may be transferred by pressure forces, in practice by horizontal pressure fluctuations on surface waves

• Heat conduction and mass transfer must take place physically through the surface, which has laminar boundary layers on each side

• Turbulence is suppressed in a laminar viscous boundary layer (thickness δ_ν ≈ 10(ν/u_∗) according to Chriss & Caldwell JGR 1984)

• The viscous boundary layer has diffusive sublayers for heat and mass whose thickness depends on the appropriate diffusion coefficient: δD ≈ δ_νSc^−1/n where the Schmidt

number Sc = ν/D, and n = 2 if the viscous and diffusion layers have the same lifetime.

Laminar layers, heat and mass flux

• During wave breaking, the air–water interface becomes discontinuous, with the

production of spray, bubbles, etc., so the effective area of the laminar regions becomes much larger, reducing the resistance to heat conduction and mass transport. Note that this does not happen in the case of ‘microbreaking’, where capillary ripples and small

overturning eddies appear in the front of a wave crest and the actual surface remains continuous.

Mechanical energy

• Mechanical energy exists in a number of different forms: energy of mean flow, wave energy, turbulent kinetic energy, and so on.

• Eventually dissipated as heat, but the thermal effects of mechanical energy dissipation may usually be neglected (except in the upper atmosphere, where the disspation of energy from internal gravity waves may cause heating by a few degrees per day).

• Typical paths:

– Energy of wind → air turbulence → heat

– Energy of wind → wind waves → ocean turbulence (via wave breaking) → heat – Energy of wind → ocean currents / internal waves → turbulence (via internal wave

breaking) → heat

• In a turbulent wall layer, turbulent kinetic energy dissipation is

ε = u_∗³/(κz) (9)

• Kolmogorov length scale L_K ≈ (ν³/ε)^1/4 may give the thickness of the laminar viscous layer: from Lorke & Peters 2006 this becomes δ_ν ≈ 10ν/u_∗.

(From Lorke & Peeters JPO 36:1591–1604, 2006)

(From Lorke & Peeters JPO 36:1591–1604, 2006)

Measurements

Many types of instrument and platform have been used, including

• Anemometers on towers, moored current meters

• Sonic anemometers, acoustic Doppler current meters

• Radiosondes, ocean profiling instruments at vertical scales from 100 m to 1 mm.

(Profiling of velocity/acceleration, pressure, temperature, composition, etc.)

• Infra-red spectrometers (for measuring composition)

• LIDAR (for aerosols etc.)

• Smoke and dye for diffusion measurements

• Buoy platforms for measuring waves, ocean and atmospheric boundary layer variables

• FLIP

• . . .

Modelling

• 3D models: isopycnic models require special treatment of the boundary layers

• Turbulence closure (turbulent kinetic energy (TKE), length scale, turbulent energy dissipation: nevertheless require a specification of the boundary conditions

• Direct numerical simulation of turbulent motions (restricted to centimetre scale)

• Large eddy simulation: appear successful in the atmospheric boundary layer, require a consistent sub-gridscale turbulence parameterisation

• Ocean mixed layer parametric models

• 1D models: may account for more processes and have a finer resolution than 3D models

• Model coupling: atmosphere–ocean, atmosphere—surface wave, atmosphere–wave–ocean, atmosphere—sea ice—ocean

Conclusions

• Many detailed observational, theoretical, and modelling studies have been made of the turbulent atmospheric and oceanic boundary layers

• In this presentation I have attempted to outline some of the concepts involved, and what may be necessary for a consistent treatment of the boundary layers on both sides of the air–water interface

• Key points:

– Turbulent wall layer behaviour may still be relevant, with modifications

– Keep track of (partially) conserved quantities: momentum, energy, mass, . . . – Laminar layers are important for heat conduction and mass transport