Effects of Turbulence

1
Effects of Turbulence

• The frictional effects on Chapter 5 were derived
  assuming a laminar or non-turbulent flow

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2
Background

• In the 1800s, Fridtjof Nansen observed that
  icebergs drifting in arctic waters did not drift
  in the direction of the local winds but rather
  with a rightward deflection to the forcing
  direction
• Contributions of friction, wind forcing
  (pressure) and Coriolis would lead to such a
  rightward deflection.
• Quantitative examination of this balance was
  proposed by Vagn Ekman in the early 1900s.

3
Turbulence is a challenging and still very open
area of study

• From Vallis
• Horace Lamb has been quoted as saying that when
  he died and went to heaven he hoped for
  enlightenment on two things, quantum
  electrodynamics and turbulence. Although he was
  only optimistic about the former.
• Quite consistently, it has been said that
  turbulence is the invention of the Devil, put on
  earth to torment us.

Geoffrey K Vallis, Atmospheric and Oceanic Fluid
Dynamics Fundamentals and Large-Scale
Circulation, Cambridge University Press, New
York, NY, 2006, pg 371
4
Turbulence

• Due to the fact that turbulent motion is
  unpredictable, the constitutive relationship
  between the stress tensor and the gradient of the
  velocity field breaks down.
• Instead we must consider a statistical approach
  by splitting a given quantity into a mean flow
  and a small random contribution
• Where

5
Turbulence

• We define the mean value as being integrated over
  a time scale, T. For example, the mean local
  acceleration is expressed as

6
Turbulence

• Recall that the equations of motion and
  conservation of mass for small density variations
  are

7
Turbulence

• Substitution of the statistical form of the
  variables of the equation of motion we obtain
  (shown without proof. For details see Chapter 9
  notes)
• What is different in this form from before?

8
The closure problem

• The inclusion of turbulent terms require
  additional equations in order to have a complete
  set of equations for the variables of interest.
• To date, it is still unclear which additional
  equations are universally acceptable to provide a
  set of consistent and unique solutions.
• One ad-hoc approach would be to represent the
  turbulent terms as a function of the mean flow .
  . .

9
Turbulent mixing constitutive hypothesis

• We want to incorporate the non-linear terms of
  the equations of motion as additional frictional
  effects due to turbulence1.
• Recall from before the stress tensor was
  comprised of a simple isotropic hydrostatic
  component and a deviatoric component
• In this case, the deviatoric tensor consists of
  all of the mean turbulent velocity components .
  i.e.

1 The paper on which most of the discussion
follows is Kirwan, A. D., Formulation of
Constitutive Equations for Large-Scale Turbulent
Mixing, Journal of Geophysical Research, 74,
6953-6959, 1969
10
Turbulent mixing constitutive hypothesis

• In this case, the deviatoric tensor is called the
  Reynolds stress tensor and consists of all of the
  mean turbulent velocity components . i.e.
• In general
• We now employ the constitutive hypothesis so that
  the Reynolds stress is proportional to the strain
  rate of the mean velocity field. (Notice we have
  to be careful about what aspect of the velocity
  field we are talking about the mean or turbulent
  component)

11
Turbulent mixing constitutive hypothesis

• The Reynolds stress tensor constitutive
  hypothesis has the mathematical form
• Where
• Now in the laminar analysis we made a set of
  assumptions about the fourth order tensor. In
  particular we assumed molecular isotropy and
  incompressibility which simplified matters
  considerably. This is physically reasonable for
  the laminar case but overly-simplified for the
  turbulent analysis.

12
Turbulent mixing constitutive hypothesis

• We will instead assume that mixing scales between
  the horizontal and vertical are quite different.
  Thus we will assume isotropy along horizontal
  planes.
• The mathematics of this analysis is beyond the
  scope of this course. Understanding of the
  underlying physical concepts is all that is
  important for now.
• Application of this analysis leads to the
  following form of the equations of motion in a
  turbulent medium

13
Turbulent equations of motion

• The equations of motion now take the form
• Where the mean overbar notation is implied and
  the laminar terms have been absorbed into the
  turbulent friction terms or simply neglected

14
Reynolds stress tensor

• Further, the Reynolds stresses are defined as

15
What is the Reynolds stress?

• Typical values of the eddy coefficients vary
  greatly
• Lower atmospheric values are
• Upper ocean values are
• Compare these with laminar flow viscosities

16
What is the Reynolds stress?

• Notice the following about the Reynolds stress
  relationship
• 1.The eddy flux is opposite to the direction of
  the shear flow. For example, for a mean flow of
  the form (shear increases in the vertical
  direction)
• The associated Reynolds stress is
• The resultant forces are in a direction opposite
  and/or lateral to the velocity field
• In analogy to laminar flow, this has a
  dissipating as well as a spreading
• or diffusive effect on the properties of the
  fluid. The following example
• may help in understanding this

17
Exercise

• For the flow field
• Find the Reynolds stresses and the direction of
  the resultant forcing

18
Exercise - Answer
So the Reynolds stress is
The direction of the stress cause the flow to
spread out or diffuse
Direction of forcing
Direction of forcing
19
Ekmans equations of motion

• Break up the flow into two contributions.
• 1. A Geostrophic flow that satisfies
• 2. A flow that balances friction with the
  Coriolis force.
• - In this case, consider a horizontally
  homogenous mixed layer, therefore no horizontal
  shear so we can neglect nh terms. Further assume
  nv is spatially constant.
• This leads to the most simple form of Ekmans
  flow equation as

20
Boundary conditions (z0)

• Consider a constant wind stress at the surface
  (z0)
• in the east-west direction.
• In terms of a constant Reynolds stress
• tensor, the wind stress has a force in the
  x-direction
• and thus takes the form
• It simplifies since vertical shear dominates
• (horizontal homogeneity assumption)

21
Boundary conditions

• We will avoid any complications with bathymetry
  by
• assuming that the flow is negligible at large
  depths.
• Therefore
• This is the same as requiring that the only
  energy
• source is due to the wind stress at the surface.
• This assumption may be relaxed later however.

22
Solving Ekmans equations

• Ekmans equation are a coupled set of 2nd order
• differential equations.
• We will focus on solving for vf first. (This
  choice is arbitrary).
• Take of equation (2) and substitute into
  equation (1).
• We obtain

23
Solving Ekmans equations

• The solution of
• Is()
• Where
• Similar analysis can show that
• There actually is a more general form of the
  solution with two more terms in each component
  of the velocity field but foresight on the nature
  of the final result allows us to simplify to the
  above for presentation purposes. The alternative
  is to impose an additional boundary condition.

24
Boundary conditions

• The requirement that
• Shows us that BD0
• And we are left with

25
Boundary conditions

• A and C are related by the fact that the
  differential equations are coupled. Substitution
  of
• into
• Shows us that and

26
Boundary conditions

• Finally, application of the stress condition at
  the surface
• allows us to solve for A. The conditions is
• Substitution of shows us

27
Solution to Ekmans equation

• We are now in a position to express a unique
  solution
• With some trigonometric manipulations, the answer
  can also
• be expressed as

28
Solution to Ekmans equation

• Now that we have a solution what does it
• tell us?

29
Solution to Ekmans equation

• We are probably familiar with the following graph
  of the solution with distance

Direction of wind forcing
Surface current
30
Solution to Ekmans equation

• 1. We can confirm the rightward deflection
  Further it is exactly 45 degrees.
• 2. Exponential decay of velocity with distance
• 3. At a depth of , the current
  direction has reversed from southeast to
  northwest. This value is arbitrarily taken as
  the effective depth of the Ekman Layer. The
  Ekman layer can be considered a length scale of
  frictional influence..

31
Are Ekman spirals observed in nature?

• Answer Only under very unique circumstances
  (Long time averages off the west US coast for
  example)
• 1. One of the motivators of assuming nv as
  spatially constant
• was simplicity rather than what actually occurs
  in nature.
• Obtaining a correct representation for nv is
  still an open
• area of research in oceanography.
• 2. Notice that our analysis only considered
  vertical
• dependence of the velocity field. Horizontal
  boundaries of
• the ocean basin should lead to horizontal
  variations in the
• ocean current as well.
• 3. There are some useful qualitative
  implications from this analysis though . . .

32
Transport and upwelling

• In general, the horizontal mass transport is
  defined
• as the horizontal momentum density integrated
  over
• a fluid column to the surface .
• The value, h, is determined by the depth of the
  mixed layer
• or where the horizontal velocity vanishes.
• MH is a measure of the amount of mass that is
  passing
• through a vertical column of the fluid per unit
  of time and
• has units of one might also relate
  mass transport to
• units of Sverdups where

33
Transport and upwelling

• We will consider an incompressible fluid and
  evaluate the
• volumetric transport of the fluid column. In
  other words we
• factor out the constant density in our
  integration.
• An westerly wind stress leads to a southward
  volume
• or mass transport of the fluid.
• In this simplified case, the mass transport of
  the ocean is at
• right angles to the wind field.

34
Transport and upwelling

• Example Bermuda high. How does the effect of
  the semi-
• permanent Bermuda high pressure system impact the
  
• Atlantic current flow along the east coast of the
  United
• States?

35
Transport and upwelling

• Example Bermuda high. How does the effect of
  the semi-
• permanent Bermuda high pressure system impact the
  
• Atlantic current flow along the east coast of the
  United
• States?
• Answer
• Southerly wind flow
• Current is deflected to East
• Incompressibility requires upwelling along US
  east coast.