Jüri Elken elken@phys.sea.ee
Tallinn University of Educational Sciences, Chair of Geophysics
Estonian Marine Institute, Department of Marine Physics
Contents
Introduction
Turbulent flow and dispersion, classical cases
Turbulence in sheared stratified flow
Turbulence description
Advection-diffusion equation
Diffusion at simple cases
Numerical models
Examples
Lecture notes by Jüri Elken Page 1

Contaminants are substances that make harm to environment and marine living organisms. In the Baltic Sea the most problematic contaminants are:
 trace metals
 organochlorines
 hydrocarbons (oil products)
 nitrogen and phosphorus causing eutrophication
Scalar tracers that may be contaminants , but also physical variables like temperature and salinity or biogeochemical variables like concentrations of oxygen, nutrients, chlorophyll and abundance of classes of phytoplankton and zooplankton
Scalar tracers may be hydrodynamically
 active - they influence the water density and currents
 passive - they have no feedback to the water dynamics
Conservative tracers like salinity and temperature undergo only advection by currents and dilution by turbulence, they have no internal sources and sinks.
Most of the tracers are non-conservative . Some of them like colibacteria undergo only destruction in time. Others like oxygen is both produced and consumed in the water column. Non-conservative tracers have sources and sinks . Many of the contaminants accumulate on the bottom and may be further released into the water column. Some non-conservative contaminants accumulate in living organism where the concentration may increase. This is called bioconcentration .
Sources of contaminants may be point sources or diffuse sources .
The source may be located as river or stream discharging contaminated water, a pipe outlet , atmospheric deposition or release from the contaminated sediments .
Lecture notes by Jüri Elken Page 2

Laminar flow is regular, layered flow that occurs at small speeds
Turbulent flow is irregular, containing of eddies that make additional friction and mix the water properties like diffusion
Reynolds found with his experiments that transition from laminar to turbulent flow takes place if
Re

UL


Re crit

2500

5000 where Re - Reynolds number, U - characteristic flow speed,
L - length scale,

- kinematic viscosity
Lecture notes by Jüri Elken Page 3

if the speed of laminar flow gets above critical then turbulence is generated that acts like friction reducing the mean flow speed
is stretched in the centre of channel where velocities are higher.
Density interface suppresses vertical mixing so that tracer remains in the upper layer where velocities are higher. Presence of velocity shear favours turbulence generation
Lecture notes by Jüri Elken Page 4

a) eddy larger than patch size makes only advective transport b) eddies comparable to patch size distort the patch: initial "red spot" is converted into "red stripes" , the eddies are called dispersive eddies, the process is called also stirring c) eddies smaller than patch size act like diffusion: initial "red spot" is converted into "pink cloud"
Deformation of "chessboard" patch in a dispersive eddy
Lecture notes by Jüri Elken Page 5

Current shear favours turbulence generation . Turbulent eddies get their energy from the mean flow. By time, turbulent eddies get smaller and smaller (this is the cascade mechanism ) until the energy is lost by friction due to molecular viscosity,
Stable density stratification suppresses turbulence generation since the kinetic energy of eddies is converted quickly to potential energy of stratification.
The balance between turbulence generation and suppression is d
 g described by Richardson number
Ri

 dz du
2 dz
, where

- water density, starts if g - gravity, u - current velocity, z - vertical axis. Turbulence
Ri

0 .
25 . This value of Ri is set also by theory of internal waves - they become unstable and start breaking .
Lecture notes by Jüri Elken Page 6

Internal waves have periods in the order from minutes to a few hours.
Krauss (1981) has shown that favourite conditions for Ri

0 .
25 appear within inertial oscillations (14 h in the Baltic) that are excited during storms.
Stratified flows are characterised also by Bulk Richardson number defined over the whole layer
Ri
B
 g u

2 h
, where h - layer thickness, u - typical velocity, g
  g

2


2

1
- reduced gravity,

1
,

2
- density of upper and lower layer.
Lecture notes by Jüri Elken Page 7

Entrainment is a process, where turbulent and laminar layers are separated by a density jump and the active turbulent layer intrudes the quiet laminar layer.
Dilution of submerged jet by entrainment is dependent on the
Froude number
Fr
 u g
 h
(ratio of current speed to phase speed of internal waves)
Fronts act as barriers to horizontal dispersion by turbulent motion and also form regions of entrapment in which dilution by vertical mixing is less effective
Lecture notes by Jüri Elken Page 8

It looks at the subsequent coordinates ( trajectories ) of a number of individual fluid or fluid property particles that are advected
(transported) by currents . When the current field is known then we may add turbulent diffusion as random walks Rnd
 
that have in general Gaussian distribution with dispersion that corresponds to turbulent activity. For a particle numbered i we get dx i dy i dz i


 u v w


 x i x x i i
,
,
, y i y i y i
,
,
, z i z z i i
,
,
, t t t


 dt dt dt



Rnd
Rnd x y
( i
 
)
Rnd z
Lecture notes by Jüri Elken Page 9

of a property

  is derived with an assumption that molecular diffusive flux is added to the advective flux
  v


 d D
 m

   d

D
. Then for incompressible fluid we obtain
 
 t
 u
 
 x
 v
 
 y
 w
 
 z
  m



 2

 x
2

 2

 y
2

 2

 z
2



P

, where q



P

is production minus destruction per unit volume and
 m

is molecular diffusion coefficient
.
Turbulent motions
are treated by
Reynolds decomposition
into a mean
, regular component and irregular pulsation
component
     
,
  
0
,
  
.
Averaging the advection-diffusion equation we obtain
 
 t
 u
 
 x
 v
 
 y
 w
 
 z
  m



 2

 x
2

 2

 y
2

 2

 z
2


  





 x u
 
 
 



 y v
 
 
 


 z w
 
 



 turbulent fluxes
, where additional "turbulent terms" appear due to non-linearity of advection.
Turbulent fluxes
are usually parameterised by gradient approach, assuming they act like diffusion
or viscosity whereas horizontal and vertical turbulence coefficients have different value u
     



 x
; v
     

 
 y
; w
     

 
 z and generally are not constants. Since turbulent diffusion is much greater than molecular, the latter can be neglected. We will reach the advection-diffusion equation of turbulent motions
 


  u

 x

 
 y

 
  v
   w

 z


 x






 x



 y






 y








 z local change advective change horizontal turbulent diffusion vertical turbulent diffusion
 net
P

 production
Lecture notes by Jüri Elken Page 10

Turbulent diffusion coefficients and current speeds are assumed constant
Vertical diffusion with no advection
 
 t
 

 2

 z
2
Instantaneous point release at the surface
If the amount M is deposited initially at the surface, then the solution is
 
M

 t exp
 z
2
4

 t
Lecture notes by Jüri Elken Page 11

Horizontal advection and diffusion
 
 t
 u
 
 x
 

 2

 x
2
Instantaneous release of substance in steady uniform flow
Initial concentration is Gaussian
 x ,

2
M
  exp


 x
2
2

2

 the solution is
 
2



2
M

2

 t
 exp

2


 x
 ut

2
2 
2

 t



Lecture notes by Jüri Elken Page 12

Steady release on a plane in a steady uniform flow diffusion across the axis of advective transport u
 
 x
 

 2
 y

2
 

 2

 z
2
For steady release at the surface Q [kg/s] the solution is
  x , y , z


4
 x
2 Q



 exp
 uy
2
4

 x
 uz
2
4

 x
Lecture notes by Jüri Elken Page 13

In realistic cases numerical models have to be used.
Continuous fields are replaced by a set of discrete values .
Time and space derivatives that appear in the equations are replaced by their discrete analogues .
There are different methods how to construct the discrete model.
Space derivatives can be taken as
 finite differences on rectangular or curvilinear grid
 finite elements on triangulated grid
Models start with initial distributions and perform time stepping .
Currents and tracers cannot have smaller structures than the model grid step .
All the eddies that size is smaller than grid step are treated as turbulent diffusion . This problem is called turbulence closure .
Diffusion coefficients are variable in space and time.
Vertical diffusion/viscosity is alternatively taken as
 function of Richardson number or other local flow parameters
 calculated from the equations for turbulent kinetic energy , called also as k
 
models
Horizontal diffusion/viscosity is alternatively taken as
 constant
 function of velocity shear (e.g. Smagorinski formulation)
Lecture notes by Jüri Elken Page 14

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Fixed levels
Sigma coordinates stretched between the bottom and surface
Lecture notes by Jüri Elken Page 17

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Lecture notes by Jüri Elken Page 19