Mixed Layer
P. N. Vinayachandran
Centre for Atmospheric and Oceanic Sciences (CAOS)
Indian Institute of Science (IISc)
Bangalore 560 012
Summer School on
Dynamics of North Indian Ocean
June-July 2010
Outline
Mixed Layer
Air-sea fluxes
Models of Mixed Layer
Bulk Models
Kraus – Turner
Price, Weller & Pinkle (PWP)
Profile Models
K-Profile Parameterizaion
Meller Yamada
Mixed Layer
Near – surface layer of the ocean. Quasi homogeneous
Properties (T, S, 
) are nearly uniform in this layer
The layer where properties change rapidly with depth is thermocline
Importance: Weather and climate (eg. cyclones)
Air – sea exchange is controlled my mixed layer processes
Life in the ocean: Most of the primary productivity occurs in the mixed layer
i) Most of the electromagnteic radiation is absorbed within the mixed layer
Ii) Mixing at the base of the mixed layer is crucial for sustaining biological productivity
How do we determine the depth of the
mxed layer
Vinayachandran et al., 2002, JGR
Difference Criterion
Properties are assumed to be uniform within the mixed layer
Isothermal, Isohaline and Isopycnal
Temperature: 0.1 to 1.0
C
Density:
( 0.2 
C)
0.01 to 0.25 kg m-3
(0.03 km m-3)
Gradient Criterion
Properties are assumed to change `sharply' just below the mixed layer
Figure courtesy: S. R. Shetye
Barrier Layer
Air – sea fluxes
Radiation
Shortwave
Downwelling radiation
Day of the year
Zenith angle
Latitude
Atmospheric transmittance (aerosol and water vapour)
Cloud cover
Upwelling shortwave radiation
Reflection from the sea surface
Albedo, Clear sky radiation, fractional cloud cover, noon solar elevation
Net Shortwave Radiation
SOC Flux data set. Josey, 1999
Penetration of short wave radiation
Solar radiation in the uppper ocean is given by
I(z) = I(0 )∑ n ez/Ln
I(0) = insolation at the surface
N = no. of spectral bands
Ln = length scale
i) shortwave
(UV < 350nm)
e-folding scale of UV ~ 5m
ii) visible
(PAR 350 – 700nm) part that penetrates
Iii) IR and nIR
(> 700nm)
absorbed in the top 10cm
I(z) = I(0) [I1 ez/L1 - I2 ez/L2]
L1 = 0.6
L2 = 20m
I1 = 0.62
I2 = 1 - I1
Paulson and Simpson, 1977, JPO
Longwave Radiation
Incoming: Radiation from the atmosphere
Outgoing: Radiation from the sea surface
Longwave
Downwelling longwave radiation
Air temperature and humidity
Upwelling longwave radiation
Reflection of longwave
radiation
Emission from the sea surface
a
Emittance of the sea surface
Vapor pressure
Cloud cover coefficient (0.5 to 0.73)
Net Longwave Radiation
Sensible heat
flux
QH = ra Cp w'T'
QH = ra Cp Ch u (Ts - Ta)
Latent heat flux
QE = ra L w 'q'
QE = ra L Ce u (qs - qa)
Net heat flux
QN = QSW + QLW + QE + QH
Annual Cycle: North Indian Ocean
Surface Buoyancy Flux
B = ro [ a (ro Cp) -1 Qnet – b So (E – P) ]
ro = surface density
So = surface salinity
a, b = Expansion coefficients for heat and salt
Cp= heat capacity
Positive buoyancy flux is stabilizing
Source: Oberhuber
(siign reversed)
Mixed layer processes
Heating by solar radiation: Penetrative
Cooling by evaporation, mechanical mixing
Heating/Cooling : Sensible heat flux/Longwave radiation
There is no echange of salt
Freshwater exchange by Evaporation and Precipitation
Changes buoyancy flux
Figure courtesy: S. R. Shetye
Figure courtesy: S. R. Shetye
Kraus – Turner Bulk Mixed Layer Model
Kraus & Turner, Tellus, 1967a: Experimental
Kraus & Turner, Tellus, 1967b: Theory
Schemtic of the mixed layer
Denman, 1973, JPO
Assumptions

Incompressible ocean.

Stably stratified fluid (Boussinesq approximation)


Wave-like dynamical effects ignored (Gravitational, inertial and Rossby
waves)
Horizontal homogenity, neglect horizontal advection
Turbulent heat flux at the base of the mixed layer
Entrainment
Integrate from z= -h to z=-h+h
0 at z=-h
In the limit as h 0
0 ; no entrainment; H=0
0 ; entrainment mixing ; H=1
Calculation of Entrainment Velocity
Production of TKE
m is an adjustable parameter; B is the buoyancy flux
1. If P > 0 ; entrainment deepening of mixed layer
2. If P < 0 ; relative importance of winds v/s convection
Monin-Obukhov Length
McCreary et al., 2001
Wind dominated regime
Wind speed = 12.5 m/s; No heat flux, No radiation
h after 24 hours = 37.1; after 48 hours = 47.8
Denman, JPO, 1973
Wind dominated regime
Run 1. Wind speed = 12.5 m/s
Run 2. Wind speed = 17 m/s
Heat dominated regime; H= 0;
Diurnal Heating
Wind speed = 4 m/s;
Profiles are plotted evey 2hrs
The PWP (Price Weller Pinkel) Model
Price et al., 1986, JGR
The (PWP) ice Weller Pinkel Model
F(0)=Q,
air-sea heat
flux
E(0)=S(E-P), fresh water flux
times
surface salinity
G(0)=
,
wind stress
Kelvin-Helmholtz
Instability
Wikepedia
Instability due to vertical shear in a stratifed fluid
Mixing
Static stability
Mixed layer stability
Shear flow stability
h
mixed layer depth difference between
mixed layer and the level just beneath.
Rb
bulk richardson number
Rg
gradient richardson number
Implementation
Vertical grid resolution, ∆z =0.5 m ; Time step, ∆t = 900 s




Begin from an initial T, S profile (eqns. 1 & 2)
Update the T, S profile for the first grid box, all the surface heat
flux goes in here
Update T profile for the boxes 2 to N using penetrative part of
radiaion
Calculate density using an equation of state, calculate mixed
layer depth for a density criterion of 0.0001

Check for static stabiity, mix and remove all instability

Calculate veloctiy profile using winds (eqn. 3)


Check for mixed layer stability, Rb > 0.65, mix and remove
instability
Check for shear flow instability, Rg > 0.25, mix and remove
instability

Rg assumed to be ≥ 0.25
If the smallest Rg < 0.25 , then
, T, S and V at the two grid
levels that produce Rg , j and j+1 are partially mixed
according to,
is the value after mixing ; Rg'
=
0.3
Rg is then recalculated from (j-1) to (j+2) and the mixing
process continues
until Rg ≥ 0.25.
KPP Mixed Layer
The turbulent mixing in the oceanic boundary layer is distinctly different from the layer below
Vertical extent of oceanic boundary layer depends on surface forcing, stratificcation and shear
Depth upto which an eddy in the boundary layer can penetrate before becoming stable.
Shallowest depth where Rb > 0.3
Monin-Obukhov Length:
Large and Gent, 1999, JPO
Below the boundary layer (interior) vertical mixing consists of:
1. Local Ri due to instability due to resolved vertical shear
2. internal wave breaking
3. double diffusion
Pacanowsky&Philander type of mixing:
KPP Vertcal Mixing
Mellor Yamada Model
Mellor, 2001, JPO
Ezer, 2000, JGR