Horizontal Mixing in Estuaries and Coastal Seas

taupeselectionΜηχανική

14 Νοε 2013 (πριν από 3 χρόνια και 6 μήνες)

63 εμφανίσεις

Horizontal Mixing in Estuaries
and Coastal Seas

Mark T. Stacey

Warnemuende

Turbulence Days

September 2011

The Tidal Whirlpool


Zimmerman (1986) examined the
mixing induced by tidal motions,
including:


Chaotic tidal stirring


Tides interacting with residual flow
eddies


Shear dispersion in the horizontal
plane


Each of these assumed
timescales long compared to the
tidal cycle


Emphasis today is on
intra
-
tidal
mixing in the horizontal plane


Intratidal

mixing may interact with
processes described by Zimmerman
to define long
-
term transport

Mixing in the Horizontal Plane


What makes analysis of
intratidal

horizontal mixing hard?


Unsteadiness and variability at a wide
range of scales in space and time


Features may not be tied to specific
bathymetric or forcing scales


Observations based on point
measurements don’t capture spatial
structure

Mixing in the Horizontal Plane


Why is it important?


To date, limited impact on modeling due to
dominance of numerical diffusion


Improved numerical methods and resolution
mean numerical diffusion can be reduced


Need to appropriately specify horizontal
mixing


Sets longitudinal dispersion (shear
dispersion)

Unaligned

Grid

Aligned

Grid

Numerical Diffusion [m
2

s
-
1
]

Holleman

et al., Submitted to IJNMF

Mixing and Stirring


Motions in horizontal plan may produce kinematic straining


Needs to be distinguished from actual (irreversible) mixing


Frequently growth of variance related to diffusivity:




Unsteady flows


Reversing shears may “undo” straining


Observed variance or second moment may diminish


Variance variability may not be sufficient to estimate mixing


Needs to be analyzed carefully to account for reversible and irreversible
mixing

Figures adapted from
Sundermeyer

and
Ledwell

(2001); Appear in
Steinbuck

et al.
in review

Candidate mechanisms for lateral mixing


Turbulent motions (dominate vertical mixing)


Lengthscale
: meters; Timescale: 10s of seconds


Shear dispersion


Lengthscale
: Basin
-
scale circulation; Timescale: Tidal or diurnal


Intermediate scale motions in horizontal plane


Lengthscales
: 10s to 100s of meters; Timescales: 10s of minutes


Wide range of scales:


Makes observational analysis challenging


Studies frequently presume particular scales


1
-
10 meters

Seconds to minutes

Basin
-
scale Circulation

Tidal and Diurnal Variations

Intermediate Scales

Turbulence

Shear Dispersion

Motions in Horizontal Plane

Turbulent Dispersion
Solutions


Simplest models assume
Fickian

dispersion


Fixed dispersion coefficient, fluxes
based on scalar
gradients


For
Fickian

model to be valid,
require scale separation


Spatially, plume scale must exceed
largest turbulent
lengthscales


Temporally
,
motions lead to both
meandering and
dispersion


Long Timescales =>
Meandering


Short Timescales =>
Dispersion



Scaling based on largest scales (dominate dispersion):


If plume scale is intermediate to range of turbulent scales, motions
of comparable scale to the plume itself will dominate dispersion



Structure of three
-
dimensional turbulence


Turbulent cascade of energy


Large scales set by mean flow
conditions (depth, e.g.)


Small scales set by molecular
viscosity


Energy conserved across
scales


Rate of energy transfer between
scales must be a constant


Dissipation Rate:

Large Scales

Intermediate

Small Scales

P

Kolmogorov

Theory


3d Turbulence


Energy
density,
E(k)
,

scaling for different
scales


Large scales:
E(k) = f(Mean flow,
e
, k)


Small scales:
E(k) = f(
e
,
n
,k)


Intermediate scales:
E(k) = f(
e
,k)



Velocity scaling


Largest scales:
u
t

= f(
U,
e

,
l
t
)


Smallest scales:
u
n

= f(
e
,
n
)


Intermediate:
u
*

= f(
e
, k)





Dispersion Scaling

k (= 1/
l)

E(k)

L.F. Richardson (~25 years prior to
Kolmogorov
)


Two
-
dimensional “turbulence” governed
by different constraints


Enstrophy

(
vorticity

squared) conserved
instead of
energy


Rate of
enstrophy

transfer constant across
scales


Transfer rate defined as
:


‘Cascade’
proceeds from smaller to
larger
scales


Two
-
dimensional
turbulent flows

Large Scales

Intermediate

Small Scales

Mean Flow

Batchelor
-
Kraichnan

Spectrum:
2d “Turbulence”


Energy density scaling changes from 3
-
d


Intermediate scales independent of mean flow, viscosity:


E(k) = f(
f

, k)




Velocity scaling


Across most scales:
u
*

= f(
f
, k)






Dispersion Scaling

k (= 1/
l)

E(k)

Solutions to
turbulent dispersion
problem


In each case,
diffusion coefficient approach leads to Gaussian cross
-
section


Differences
between solutions can be described by the lateral extent or variance
(
s
2
):





Constant diffusivity solution




Three
-
dimensional scale
-
dependent solution





Two
-
dimensional scale
-
dependent solution


t
K
b
x
y
2
)
(
2
2


s
b
Ut
e
b
x
/
2
2
2
)
(

s

y
y
K
t
2
2



s

)
constant

y
K

)
3
/
4
3
/
1
s
e

y
K

)
2
3
/
1
s
f

y
K
3
2
2
3
2
1
)
(








b
Ut
b
x

s
Okubo Dispersion Diagrams


Okubo (1971) assembled historical data
to consider lateral diffusion in the ocean


Found variance grew as time cubed within
studies


Consistent with diffusion coefficient
growing as scale to the 4/3

Shear Dispersion


Taylor (1953) analyzed dispersive effects of vertical shear
interacting with vertical mixing


Analysis assumed complete mixing over a finite cross
-
section


Unsteadiness in lateral means Taylor limit will not be
reached


Effective shear dispersion coefficient evolving as plume grows and
experiences more shear


Will be reduced in presence of unsteadiness

l
z

l
y

Developing Shear Dispersion


Taylor Dispersion assumes complete mixing over a vertical
dimension, H, with a scale for the velocity shear, U:



Non
-
Taylor limit means H =
l
z
(t):



Assume locally linear velocity profile:



Velocity difference across patch is:



Assembling this into Taylor
-
like dispersion coefficient:



z
Taylor
K
H
U
K
2
2


)
t
K
t
z
z
2
0


l
l
z
U
z
U



0
)
(

)
t
K
t
U
z
2
0

l

l





2
2
2
2
2
2
2
4
4
t
K
K
t
K
K
U
K
z
z
z
z
z
y


l




3
2
2
2
3
4
2
t
K
K
t
z
y

s
s





Okubo Dispersion Diagrams


Okubo (1971) assembled historical data
to consider lateral diffusion in the ocean


Found variance grew as time cubed within
studies


Consistent with diffusion coefficient
growing as scale to the 4/3

Horizontal Planar Motions


Motions in the horizontal plane at scales intermediate to
turbulence and large
-
scale shear may contribute to
horizontal dispersion


Determinant of relative motion, could be dispersive or ‘anti
-
dispersive’ (i.e., reducing the variance of the distribution in the
horizontal plan)

Framework for Analyzing Relative Motion


In a reference frame moving at the velocity of the center
of mass of a cluster of fluid parcels, the motion of
individual parcels is defined by:





Where (
x,y
) is the position relative to the center of mass


Relative motion best analyzed with
Lagrangian

data


For a fixed
Eulerian

array, calculation of the local velocity
gradients provide a snapshot of the relative motions experienced
by fluid parcels within the array domain


































y
x
y
v
x
v
y
u
x
u
v
u
Structures of Relative Flow


Eigenvalues

of velocity gradient tensor determine relative
motion: nodes, saddle points, spirals, vortices


Real
Eigenvalues

mean nodal flows:

Stable Node:

Negative
Eigenvalues

Unstable Node:

Positive
Eigenvalues

Saddle Point:

One Positive, One Negative

Structures of Relative Flow


Eigenvalues

of velocity gradient tensor determine
relative motion: nodes, saddle points, spirals, vortices


Complex
Eigenvalues

mean vortex flows:

Stable Spiral:

Negative Real Parts

Unstable Spiral:

Positive Real Parts

Vortex:

Real Part = 0

Categorizing Horizontal Flow Structures


Eigenvalues

of velocity gradient tensor analyzed by Okubo
(1970) by defining new variables:







With these definitions,
eigenvalues

are:


Dynamics

g

Okubo, DSR 1970


Categorization of flow
structures can be reduced to
two quantities:


g

determines real part



determines
real v. complex


Relationship between and
g

differentiates nodes and saddle
points


Time variability of ,
g

can be
used to understand shifting
fields of relative motion

Implications for Mixing


Kinematic straining should be
separated from irreversible
mixing


Flow structures themselves may be
connected to irreversible mixing


Specific structures


Saddle point: Organize particles
into a line, forming a front


Anti
-
dispersive on short timescales,
but may create opportunity for
extensive mixing events through
folding


Vortex: Retain particles within a
distinct water volume, restricting
mixing


Isolated water volumes may be
transported extensively in
horizontal plane


McCabe et al. 2006

Summary of theoretical background


Three candidate mechanisms for lateral mixing, each
characterized by different scales


Turbulent dispersion


Anisotropy of motions, possibly approaching two
-
dimensional
“turbulence”


Wide range of scales means scale
-
dependent dispersion


Shear dispersion


Timescale may imply Taylor limit not reached


Unsteadiness in lateral circulation important


Horizontal Planar Flows


Shear instabilities, Folding, Vortex Translation


May inhibit mixing or accentuate it

Case Study
I
: Lateral Dispersion in the BBL


Study of plume structure in
coastal
BBL (Duck, NC)


Passive, near
-
bed, steady dye
release


Gentle topography


Plume dispersion mapped by
AUV

Plume mapping results


Centerline concentration and plume width vs. downstream distance


Fit with general solution with exponent in scale
-
dependency (n) as tunable
parameter















n=1.5 implies energy density with exponent of
-
2

n= 1.5

n= 1.5

Compound Dispersion Modeling


As plume develops, different dispersion models are
appropriate


4/3
-
law in near
-
field; scale
-
squared in far
-
field


4/3
-
law

Scale
-
squared

Compound Analysis

Actual Origin

Virtual Origin

Matching

Condition

Compound Solution,
Plume
Development


Plume scale smaller than largest
turbulent scales


Richardson model (4/3
-
law) for
rate of growth


Meandering driven by largest 3
-
d
motions and 2
-
d motions


Plume larger than 3
-
d turbulence,
smaller than 2
-
d


Dispersion Fickian, based on
largest 3
-
d motions


2
-
d turbulence defines
meandering


Plume scale within range of 2
-
d
motions


2
-
d turbulence dominates both
meandering and dispersion


Rate of growth based on scale
-
squared formulation

Spydell

and
Feddersen

2009


Dye dispersion in the coastal zone


Contributions from waves and wave
-
induced currents


Analysis of variance growth


Fickian

dispersion would lead to
variance growing linearly in time


More rapid variance growth attributed
to scale
-
dependent dispersion in two
dimensions


Initial stages, variance grows as
time
-
squared


Reaches
Fickian

limit after several
hundred seconds

Jones et al. 2008


Analysis of centerline
concentration and
lateral scale


Dispersion coefficient
increases with scale to
1.23 power


Consistent with 4/3 law
of Richardson and
Okubo


Coefficient 4
-
8 times
larger than Fong/Stacey,
likely due to increased
wave influence

Dye, Drifters and Arrays


Each of these studies
relied on dye dispersion


Limited measurement of
spatial variability of velocity
field


Analysis of motions in
horizontal plane require
velocity gradients


Drifters:
Lagrangian

approach


Dense Instrument arrays
provide
Eulerian

alternative

Summary of Case Study
I


Scale dependent dispersion evident in coastal bottom
boundary layer


Initially, 4/3
-
law based on three
-
dimensional turbulent structure
appropriate


As plume grows, dispersion transitions
to
Fickian

or exponential


Depends on details of velocity spectra



Dye Analysis does not account for kinematics of local velocity
gradients


Future opportunity lies in integration of dye, drifters and fixed moorings



Key
Unknowns:


What is the best description of the spectrum of velocity fluctuations in
the coastal ocean? What are the implications for lateral dispersion
?


What role do intermediate
-
scale velocity gradients play in coastal
dispersion?


How should scalar (or particle) dispersion be modeled in the coastal
ocean? Is a
Lagrangian

approach necessary, or can traditional
Eulerian

approaches be modified to account for scale
-
dependent dispersion?

Recent Studies II: Shoal
-
Channel Estuary


Shoal
-
channel estuary provides environment to study
effects of lateral shear and lateral circulation


Decompose lateral mixing and examine candidate mechanisms


Pursue direct analysis of horizontal mixing coefficient


Shoal

Channel

All work presented in this section from
:
Collignon

and Stacey, submitted to JPO,

2011

Study site


ADCPs at channel/slope, ADVs on Shoals, CTDs at all


Boat
-
mounted transects along A
-
B
-
C line


ADCP and CTD profiles

A

B

C

A

B

C

channel

slope

shoal

Decelerating Ebb, Along
-
channel Velocity

Colorscale
:
-
1 to 1 m/s

T4

T6

T8

T10

Salinity

T6

T8

T10

T4

Colorscale

23
-
27
ppt

Cross
-
channel velocity

T6

T8

T10

T4

Colorscale
:
-
.2 to .2 m/s

Lateral mixing analysis


Interested in defining the net lateral transfer of momentum
between channel and shoal


Horizontal mixing coefficients


Start from analysis of evolution of lateral shear:

Dynamics of lateral shear

Convergences and
divergences intensify
or relax gradients

Longitudinal
Straining

Variation in
bed stress

Lateral mixing

Each term calculated from March 9 transect data
except lateral mixing term, which is calculated as the
residual of the other terms

Bed
StressTerm

Time

Lateral position

Depth

Term
-
by
-
term Decomposition

inferred

Ebb

Floo
d

Time
[day]

channel

slope

shoal

Ebb

Flood

Time [day]

Convergences and lateral structure


Convergence evident in late ebb


Intensifies shear, will be found to compress mixing

POSITION ACROSS INTERFACE

POSITION ACROSS INTERFACE

ACROSS CHANNEL VELOCITY

ALONG CHANNEL VELOCITY

Term
-
by
-
term Decomposition

inferred

Ebb

Floo
d

Time
[day]

channel

slope

shoal

Lateral eddy viscosity: estimate

From Collignon and Stacey (2011), under review,
J. Phys. Oceanogr.

Linear fit

Background:

Contours:

Ebb

Flood

channel

slope

shoal

Inferred mixing coefficient


Inferred viscosities around 10
-
20 m
2
/s


Turbulence scaling based on tidal velocity and depth less than
0.1 m
2
/s


Observed viscosity must be due to larger
-
scale mechanisms


Lateral Shear Dispersion Analysis

v [m/s]

s [psu]

Lateral Circulation over slope consists of exchange flows but with large
intratidal

variation

Repeatability

Depth
-
averaged longitudinal vorticity ω
x

measurements from
the slope moorings show similar variability during other
partially
-
stratified spring ebb tides


<
ω
x

> [s
-
1
]

Lateral circulation

ω
x

> 0

ω
x

> 0

ω
x

< 0

2
nd

circulation reversal (late
ebb): driven by lateral density
gradient,
Coriolis
, advection

1
st

circulation reversal (mid ebb):
driven by lateral density
gradient
induced by spatially variable mixing

Implications of lateral circ for dispersion


Interaction of unsteady shear and vertical mixing


Estimate of vertical diffusivity:



Mixing time:



Circulation reversals on similar timescales


Taylor dispersion estimate:



Would be further reduced, however, by reversing, unsteady, shears

1.3 hours

1.5 hours

Horizontal Shear Layers


Basak

and
Sarkar

(2006)
simulated horizontal shear
layer with vertical
stratification

Horizontal eddies of vertical
vorticity

create density perturbations and mixing

Lateral Shear Instabilities


Consistent source of shear due to
variations in bed friction


Inflection point and
Fjortoft

criteria for
instability essentially always met


Development of lateral shear
instabilities limited by:


Friction at bed


Timescale for development

Lateral eddy viscosity: scaling

From
Collignon

and Stacey (2011), under review,
J. Phys.
Oceanogr
.

Mixing length scaling based on large
scale flow properties



Characteristic velocity:



Mixing length: vorticity thickness





Linear fit:


Estimate (o)

Scaling (+)

Effect of convergence front

Flood

Ebb

Implications for Lateral Mixing

Fischer (1979)

Measurements in
unstratified channel flow:



Basak

&
Sarkar

(2006)

DNS
of
stratified
flow with
lateral shear:



Bottom generated turbulence

Shear instabilities

Observations show that lateral mixing at the shoal
-
channel
interface is dominated by lateral shear instabilities rather than
bottom
-
generated turbulence.

Summary: Case Study II


Lateral mixing in shoal
-
channel
estuary likely due to combination
of mechanisms


Shear dispersion due to exchange
flow at bathymetric slope


Lateral shear instabilities


Intratidal

variability fundamental to
lateral mixing dynamics


Exchange flows vary with
timescales of 10s of minutes


Lateral shear instabilities


Horizontal scale of 100s of meters,
timescales of 10s of minutes


Convergence fronts alter effective
lengthscale


Key Unknown: What is relative
contribution of intermediate scale
motions in non
-
shoal
-
channel
estuaries


Intermediate scales appear to
dominate in shoal
-
channel system

Summary and Future Opportunities


Lateral mixing in coastal ocean appears to be
characterized by scale
-
dependent dispersion processes


Could be result of turbulence or intermediate scale motions


Estuarine mixing in horizontal plane due to combination of
lateral shear dispersion and intermediate scale motions


Intratidal

variability fundamental to mixing process


Creates particular tidal phasing for lateral exchanges


Future Opportunities:


Clear delineation of anisotropy in stratified coastal flows and
associated velocity spectra/structure


Role of bathymetry in establishing lateral mixing processes


Parameterization for numerical models



Thanks!


Contributors:
Audric

Collignon
,
Rusty
Holleman
, Derek Fong


Funding: NSF (OCE
-
0751970,
OCE
-
0926738), California
Coastal Conservancy


Special Thanks to Akira Okubo
for figuring this all out long
ago…