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From YouTube: Andrew Shao 2020 05 11
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B
So
yeah
so
I
want
to
just
give
a
talk
today
about
how
we
do
all
neutral
diffusion
in
mom
six.
So
this
was
a
project
that
was
what
I
did
when
I
was
at
kfdl,
and
you
know
I
much
thanks
and
appreciation
to
Bob,
Alistair
and
Steve
for
kind
of
taking
me
under
the
wing
and
trying
to
help
me
understand
the
problem
and
Implement
something
new.
B
So
some
of
this
Builds
on
some
of
the
material
that
we've
been
talking
about,
oddly
enough
with
the
Rewritten
and
remapping
part
of
it,
where
we,
where
you
apply
some
of
those
same
Concepts
and
ideas
to
this
problem
of
horizontal
diffusion.
So
hopefully
there.
This
was
kind
of
one
of
the
ways
that
I
figured
out
how
the
regrading
or
mapping
was
going
on.
So
part
of
this
is
going
to
be
my
exposition
of
what
it
means
to
do
a
reconstruction
in
the
vertical
and
do
some
application
of
that.
B
B
So
one
of
the
things,
though,
that
we
need
to
be
careful
of
when
we
talk
about
what
a
reaction
parameterizing
is
the
effect
of
is,
is
whether
we're
talking
about
mixing
whether
we're
talking
of
stirring
what
you
know
are
we
talking
about
enhanced
viscosity?
What
what
we're
talking
about
so
for
the
problem
of
neutral
diffusion
or
the
kind
of
lateral
or
horizontal
diffusion
in
general?
B
What
we're
talking
about
are
the
the
this
idea
of
an
Eddy
deforming,
a
tracer
field,
so
there's
the
picture
or
the
diagram
of
the
right
is
very
famous
One
I.
Think
at
this
point
it's
65
years
old,
but
it's
still
one
of
I
think
conceptually
one
of
the
easiest
ways
that
I
think
of
why
we
think
that
Eddies
might
actually
contribute
to
enhanced
diffusion
so
shown
on
the
right
hand.
Side
is
just
this
is
a
low
pressure
Eddy
and
there's
a
checkerboard
pattern.
B
That's
just
been
superimposed
onto
that
and
the
evolution
of
this
checkerboard
pattern
as
it
gets
affected
around
in
this
in
this
flow
field,
that's
caused
by
this
Eddy.
You
can
see
that
it
begins
to
stretch
out
right.
So
it
goes
from
this
nice
Square
checkerboard.
It
forms
deforms
and
deforms.
B
It
starts
to
filament,
so
that
filamentation
tends
to
it
actually
sharpens
the
Tracer
gradients
right,
and
so,
when
you
sharpen
the
Tracer
gradient,
then
you
can
actually
get
a
larger
flux
because
molecular
at
the
molecular
level
because
you
have
a
sharper
gradient,
and
so
this
is
kind
of
a
more
formalized
by
the
Boost
Nest
hypothesis
adapted
for
tracers.
The
Boost
hypothesis
originally
was
intended
more
for
viscosity
in
the
mixing
momentum
by
by
turbulence,
but
you
can
essentially
it
essentially
says
that
you
can
effect
model
the
effect
of
turbulence
as
an
enhanced
downgradient
diffusion.
B
So
that's
kind
of
the
thing
that
we're
talking
about
today,
specifically
in
the
ocean.
This
mixing
happens
along
the
direction
of
neutral
buoyancy
right,
so
we
know
that
processes
in
the
ocean
tend
to
be
adiabatic,
and
so
originally
they
were
thinking
was.
Is
that
mixing
happens
along
an
isopycnal
Direction
later
on?
There
was
a
there's,
been
a
lot
of
literature.
That's
talking
about.
Okay,
the
isopytonal
is
not.
C
B
Neutral
in
the
sense
of
their
that
true
neutrality
means
that
there'd
be
no
buoyancy,
flux
I'll,
get
into
that
a
little
bit
later.
B
That
distinction,
but
but
for
now
just
think
about
the
idea
that
we
we
don't
want
to
diffuse
along
the
surfaces
of
the
model,
which
can
be
as
as
Alistair
Steve
and
Bob
were
talking
about
before
we
have
a
gender
mom6
is
a
general
coordinate
model,
so
we're
not
guaranteed
that
the
surfaces
of
the
model
layer,
space
of
the
model
is
isopypnol
or
not
so,
and
especially
for
a
model
like
pop
or
others
as
star
models
or
Z
level
models
or
Sigma
models.
B
You
have
to
do
some
type
of
rotation
if
you
want
to
orient
the
diffusive
fluxes
along
an
isopycnel,
so
this
has
been
done
for
quite
a
long
time.
One
of
the
things
that
this
is
also
called
is
ready
type
diffusion,
and
that
was
because
ready
back
in
1982
she
kind
of
formalized
the
tensor
form
of
the
of
the
algebra
and
and
and
showed
a
way
that
you
could
calculate
the
fluxes
by
rotating
the
diffusion
tensor
along
these
isotopical
directions.
B
So
from
the
core
to
the
model,
rotate
the
diffusion
tensor,
so
that
you
have
not
just
horizontal
diffusive
fluxes
but
also
vertical
fluxes,
and
that
those
two
together
are
oriented
along
an
isopypnol
plane.
Cox
1987
showed
that
in
a
globe
model
that
this
generally
improved
model
scale,
but
they
still
needed
background
horizontal
diffusion
to
do
some
kind
type
of
no
mixing
and
later
on.
This
was
kind
of
recognized
that
that.
B
This
background,
horizontal
distribution
was
done
away
with
by
John
McWilliams,
which
is
a
different
part
of
a
tracer
effect.
So
in
griffisol,
98
came
along
and
proposed
a
way
to
calculate
like
this,
so
that
there
was
no
net
buoyancy
flux
and
to
discretize
the
operator
itself
to
ensure
that
Tracer
variance
reduces
everywhere
right
so
diffuse.
This
is
what
Tracer,
what
diffusion
tends
to
do
right
you?
B
You
never
want
to
sharpen
a
gradient,
it
always
diffuses
gradients,
which
leads
to
a
production
and
Tracer
variance,
and
then
there's
been
a
series
of
papers,
probably
one
of
the
more
recent
ones
that
show
that
the
way
that
you
calculate
slopes
typically,
what
was
done
in
these
implementations
was
that
they
use
a
local
estimate
of
what
the
isopipital
slope
was
or
the
neutral
tangent
plane,
but
gross
camp
and
all
really
identify
a
nice
way
in
a
nice
way
to
show
that
you
can
actually
get
an
accuracy
with
that.
B
So
the
equation
on
the
right
hand,
side,
you
know,
there's
just
some
some
basic
basic
terminology
and
notation.
So
this
operator,
this
tensor
operator
capture
row,
is
just
the
along
isopic
multi
diffusion
operator.
The
next
set
of
equations
is
kappa's,
small
is
with
what
happens
if
you
assume
that
isopyptus
slopes
are
are,
are
small
or
neutral,
planes
are
small,
the
last
one
is
probably
the
one
that
is
actually
much
more
germane
to
the
conversation
that
we're
talking
about
here.
B
Where
we're
essentially
saying
okay,
you
have
a
diffusion
of
temperature
and
you
have
a
diffusion
of
salinity.
So
if
you
compare
the
buoyancy
gain
or
loss
by
those
by
multiplying
them
by
the
thermal,
the
thermal
expansion
coefficient
and
the
Halon
contraction
coefficient
respectively,
you
have
you
generate
a
necessary
condition
for
a
no
net
buoyancy
gain
or
loss
due
to
the
diffusion
of
temperature
and
salinity.
B
B
So
there
were
a
couple:
a
couple
of
papers
in
1998,
including
the
Griffin's
paper
that
I
mentioned
that
showed
some
problems
where
the
rotated
tensor
approach,
so
one
in
particular
showed
belt
blockers
at
all
in
1998
showed
in
particularly
that
no
local
rotated
tensor
scheme
can
be
guaranteed
to
be
positive
definite.
So
this
actually
is
a
big
problem,
because
it
means
that
you
can
actually
get
new
extrema
in
your
Tracer
field.
B
If
you
try
to
do
this
rotated
approach-
and
we
know
that
you
know-
diffusion
like
I
said
tends
to
wants
to
reduce
trace
of
variance
Griffin's
1998
also
demonstrated
that
the
original
operating
operator
is
also
susceptible
to
instability.
B
So
if
you
look
on
the
figure
on
the
right
hand,
side,
this
sensibility
shows
up
after
I
think
something
like
a
thousand
years
of
integration,
but
you
can
see
that
instead
of
having
a
nice
smooth
thermal
client,
you
actually
get
these
Jags
and
in
the
northern
part
of
the
Basin
around
60
degrees
north
in
that
top
panel.
He
also,
he
also
like
I,
said
made
some
good
thermodynamic
arguments
about
how
you
can
actually
calculate
explicitly
a
no
net
buoyancy
loss,
and
so
that's
so.
B
His
modifications
to
the
algorithm
are
shown
in
the
bottom
panel
there
and
you
can
see
that,
especially
in
the
60
degrees
north,
where
you
have
these
strong,
strong,
deep
nice
picnos
that
you
get
a
much
smoother
thermocline,
as
opposed
to
the
top
panel,
where
you
have
this
kind
of
noise
that
within
the
thermocline
one
key
other
point,
is
that
when
you're
calculating
a
slope
right,
a
slope
is
just
simply
a
change
over
a
horizontal
distance
over
a
change
in
a
vertical
distance.
So
what
happens
then?
B
If
you
have
very
unstratified
regions,
you
get
very
large
slopes,
and
so,
for
numerical
reasons
you
need
those
slopes
need
to
be
tapered
and
limited.
So
this
happens
near
the
boundary
layers
quite
a
lot,
or
it
can
also
happen
in
regions
where
the
ocean
is
weakly
stratified,
like
the
Southern
Ocean
parts
of
the
Arctic
Ocean.
B
So
so
it's
something
that's
necessary
due
to
poor
numerical
reasons,
but
it
has
no
physical
justification
whatsoever
and
so
there's
a
nice
paper
by
a
non-get
on
a
desk
in
in
2007
that
showed
that
there's
actually
some
just
this
purely
empirical,
not
physics-based
Choice,
can
actually
lead
to
changes
in
precipitation
and
changes
in
the
vertical
structure,
feed
uptake.
So
it's
a
climate
scale
response
to
something:
that's
completely
arbitrary.
B
So
for
those
reasons
we
wanted
to
come
up
with
the
way
I
doing
something
completely
different
from
a
rotated
operator
approach
that
solves
some
of
these
problems.
So
the
main
goals
of
this
is
to
create
a
new
discretized,
epineutral
diffusion
operator,
epineutical,
meaning
being
a
long
and
neutral
direction
that
is
suitable
for
a
general
coordinate
model.
B
So
it
can
work
in
any
arbitrary
coordinate
that
preserves
the
extrema
so
that
you
don't
get
new
new
new
values,
new
maximum
new
minimum,
new
Maxima
in
your
field
or
Minima,
and
it
has
no
need
for
this
regularization
or
tapering
at
the
slopes.
B
B
So
that's
a
very
dense,
very
jargony
kind
of
summary
of
it,
and
so
my
goal
for
at
the
end
of
this
presentation
is
that
we
can
at
least
that
will
unpack
some
of
these
specific
jargony
kind
of
things
and
hopefully
give
you
a
nice
idea
of
visual
idea
or
conceptual
idea
of
of
what
we
mean
by
the
statement
right
here.
B
So
just
a
quick
note
and
an
advertisement
for
the
next
series
of
talks,
so
every
neutral
diffusion
is
appropriate
for
the
adiabatic
interior
ocean.
So
there
are
some
arguments
that
we
made
that
our
eddieplexes
are
not
actually
epidem
neutral
in
the
boundary
layers.
That's
both
in
the
service
boundary
layer
or
near
the
bottom
boundary
layer,
so
Gustavo
Marquez
is
going
to
be
talking
in
the
next
set
of
discussions
for
how
to
deal
with
this
for
how
to
deal
with
diffusion
at
the
surface
and
bottom
boundulators.
B
Okay,
so
Bob
ellister
and
Steve
spent
a
little
bit
of
time
talking
about
polynomial
reconstructions
that
underpin
the
re-gridding
and
the
remapping
that
happened
in
mom
six,
so
for
here
this
is
the
way
that
I
or
that
have
really
helped
me
visualize
it.
So
we
start
from
recognizing
that
a
tracer
concentration
in
a
finite
volume
model,
like
Mom
6,
represents
just
a
cell
average
and
represents
a
cell
average
right.
It's
a
single
concentration
that
spans
the
entirety
of
that
finite
thickness
that
that
a
layer
contains
in
the
model.
B
So
when
we
talk
about
these
kind
of
polynomial
reconstructions
we're
talking
about
generating
something
that's
in
between
those
things.
So
if
you
looked
at
a
purely
in
layer
space
and
just
took
that
this,
that
that
statement,
that
a
tracer
concentration
is
a
cell
average
and
a
vertical
distributation,
you
have
what
we
call
a
piecewise,
constant
representation
of
a
tracer
in
the
vertical.
B
So
that's
kind
of
shown
here
on
the
leftmost
panel,
the
leftmost
column,
where,
if
you
just
plot
that
temperature,
then
right
you
get
these
very
large
jumps
where
you
have
just
the
changes
in
the
layer
right
and
that
represents
actually
a
physical
system
that
is
full
of
hydraulic
shocks
and,
as
we
know,
especially
in
the
quiet,
some
deeper
part
of
the
ocean,
the
ocean
is
not
hydraulic.
Does
not
have
these
big
hydraulic
jumps.
B
So
a
second
order,
polynomial
something
like
x,
squared
plus
you
know,
plus
a
times
X,
plus
a
constant,
and
that
gives
you
a
parabolic
reconstruction
and
so
that's,
what's
kind
of
represented
on
the
third
panel
there,
where
you
now
have
this
much
smoother,
looking
profile
that
if
you
were
to
that,
looks
much
more
like
something
that
you
would
actually
measure
with
a
ctd
in
the
actual
ocean
right,
so
something
that
just
smoothly
varies
in
the
vertical,
as
opposed
to
something
that
like,
in
contrast
to
something
a
left-hand
side
which
is
very,
very
jumpy.
B
So
there's
a
couple
of
terminologies
I.
Don't
really
want
to
go
into,
but
we
want
these
things
to
be
monotonic,
so
that
they're
always
increasing
or
decreasing
with
themselves,
and
that
these
also
don't
introduce
any
new
extrema.
B
So
and
then
one
last
point
is
that
we
call
these
kind
of
discontinuous
reconstructions
because,
as
you
can
see,
especially
in
the
linear
example,
you
can
still
have
a
jump
at
the
interface
between
layers,
and
so
that's
what
we
call
a
discontinuity
so
just
to
briefly
describe
the
algorithm
of
the
whole.
But
there
are
three
different
phases
of
our
neutral
diffusion
algorithm
one.
B
We
we
generate
these
polynomial
reconstructions
for
every
Tracer,
but
particularly
for
temperature
and
salinity
for
some
of
the
examples
I'm
showing
here
they're
going
to
be
piecewise,
linear,
just
for
the
sake
of
explaining,
because
I
don't
know
about
you,
but
I
can't
interpolate
on
a
parabola
easily,
but
I
can
interpolate
online
much
more
easily.
B
And
then
the
next
thing
that
we
do
in
this
initialization
stage
is
that
we
calculate
these,
the
thermohaline
or
the
thermal
expansion
and
haline
contraction
coefficients
at
the
model
interfaces
and
by
a
model
interface.
What
I
mean
is
that,
if
you're
thinking
about
this
in
terms
of
a
case
base
and
using
this
one
half
indexing
an
interface,
is
that
intercept?
Is
that
point
where
a
layer,
the
model
meets
another
layer
of
the
model?
B
B
We
also
filter
out
the
unstable
parts
of
the
water
column,
I'm
not
going
to
go
into
that
too
much
right
now,
since
I
think
it
just
it's
a
it's
a
it's
something
that
we
need
to
do,
but
it
might
obfuscate
the
explanation
the
algorithm
a
little
bit,
but
if
anybody's
curious
about
why
we
do
that
feel
free
to
follow
up,
then
there's
a
sorting.
B
What
we
call
a
sorting
phase
and
this
sorting
phase
is
when
we
actually
create
some
sub
layers
of
the
model
that
are
bounded
by
Surfers
that
are
mutually
buoyant
and
again,
like
I'll,
spend
some
time
talking
about
that
in
the
next
couple
slides.
So
if
you
don't
immediately
understand
what
we
mean
by
that,
hopefully
be
made
clear
and
then
the
last
part
of
the
algorithm
is
when
we
actually
calculate
the
Tracer
fluxes,
Within
These
sub
layers,
I
mean
we
have
to
do
some
limiting
to
to
essentially
maintain
consistency
with
the
algorithm
itself.
B
So
let's
go
jumping
so
real
quick,
then
just
point
the
figure,
the
interfaces,
the
vertical
interface
of
the
model
are
represented
by
this
K
minus
three
hack,
one
half
plus
one
half
and
three
halves
and
then
what's
shown
is
the
blue
dub.
The
blue
line
is
what
say:
a
potential
density
would
look
like
within
this
kind
of
hypothetical
situation
based
on
the
polynomial
reconstructions
of
temperature
and
salinity.
B
You
know
for
this
particular
case.
The
density
profile
is
monotonic
in
potential
density,
space
so
and
I'm
using
the
sigma
notation
here.
Just
to
represent
potential
density,
which
is
another
simplification
that
I'm
making
for
the
purpose
of
explaining
the
talk
I'm
in
The
Zeta
coordinate,
is
just
a
non-dimensional
vertical
Port
vertical
variable
vertical
coordinate
variable
that
has
zero
at
the
top
of
a
layer
and
one
at
the
bottom
of
the
layer.
B
Okay.
So
when
we
talk
about
neutral
density
or
talking
about
the
difference
in
density,
this
is
the
equation,
essentially
that
we're
evaluating
right
this
difference
in
dense
date
is
equal.
Just
to
the
difference
in
is
equal
to
the
difference
in
densities,
and
if
you
expand
that
in
terms
of
the
derivatives
of
the
equation
or
state,
what
you
get
is
that
it's,
the
diff
it's
the
average
of
the
thermal
expansion
coefficients
times
the
great
times
the
difference
in
temperature,
plus
the
average
of
the
Halon
contraction
coefficients
times
the
difference
in
salinity
right.
So
this
will
so.
B
This
admits
a
lot
of
different
things
in
the
sense
that
two
water
particles
can
have
completely
difference:
temperature
and
salinities,
but
just
by
virtue
of
the
fact
that
you
have
a
this,
you
know
somewhat
some
different
values
for
these.
For
the
thermal
expansion
handling
contraction
coefficients,
you
can
actually
still
have
those
two
water
particles
be
neutrally
buoyant.
B
So
if
you
use
the
midpoint
pressure
or
the
average
pressure
between
the
two
wire
Parcels
Institute
pressure,
you
get
basically
a
neutral
density.
This
represents
a
difference
in
neutral
density.
If
you
use
a
single
reference
pressure,
for
example,
like
Sigma
2,
where
everything
is
referenced
to
2000
decibar,
you
get
a
different
Delta
row,
then,
is
just
simply
an
isopygninal
in
that
sense,
so
to
find
the
surfaces
of
neutral
density,
we
we
start
from
the
from
the
top
of
the
water
column
and
then
conceptually
search
down
so.
A
B
At
the
densities
at
the
top
at
the
top
interface
first,
so
if
you
think
that
so,
for
example,
here
we
have
density
increasing
and
the
scale
of
this
is
kind
of
that,
the
further
the
closer
you
are
to
the
right,
the
denser,
your
water
column
is
so
the
left-hand
column
is
actually
lighter
than
the
right
hand,
column,
so
we're
searching
down,
and
we
know
that
this
top
interface
on
the
left
hand
side
is
lighter
than
anything
is
lighter
than
the
top
interface
on
the
right
hand
side.
B
So
that
does
that
holds
true
for
all
these
cases,
except
for
the
interface
at
the
left
hand,
side,
that's
bounded
by
K,
plus
one
half
and
K,
plus
three
halves
where
we
can
see
that
this
density
profile
it
crosses
this,
it
crosses
the
it
has
the
same
density
somewhere
within
that
cell.
So
this
is.
B
This
is
what
we
mean
by
sorting
right,
we're
just
kind
of
going
down
we're
finding
these
places
where
the
two
water
columns
over
one
lap,
and
so
this
one
interface
on
the
right
hand,
side
this
first
red
line
with
an
arrowhead
on
it.
You
can
see
that
that
is
where
the
two
densities,
the
two
isoprignal
densities,
are
exactly
the
same
right.
B
We
always
search
from
an
interface.
We
never
search
anywhere
else
starting
from
it,
but
we
all
we
can
connect
it
somewhere
on
the
other
part
of
the
column
somewhere
in
between
that
layer
right
and
then
so
we,
and
so
likewise,
then,
for
the
subsequent
Red
Line
this
next
Red
Line,
then
you
have
it
pointing
from
an
interface
in
this
case
it
points
from
K
plus
three
halves
on
the
left,
hand,
column
and
points
to
the
right
part
of
it
right.
B
The
the
place
in
the
layer
of
the
right
hand
column
where
the
density
is
equal.
So
now
we
have
the
sub
layer
of
the
model
right
that
that
is
bounded
by
these
surfaces,
where
Delta
rho
is
equal
to
zero
that
have
a
thickness
on
both
sides
of
it.
So
this
is
what
we're
calling
a
sub
layer
of
the
model.
So
it's
a
layer.
So
it
has.
You
have
these
interfaces
that
are:
are
these
kind
of
generated
interfaces
represented
by
the
red
lines
that
bound
a
layer,
a
sub-layer
of
finite
thickness?
B
So
just
a
couple
of
points
when,
if
you
look
at
this
equation,
this
is
essentially
equivalent
to
finding
the
root
of
a
high
order
polynomial.
So
you
have
to
do
something
when
a
layer
should
connect
within
it,
and
there
are
also
so
this
is.
It
essentially
boils
down
to
root
finding
for
a
higher
to
polynomial,
there's
a
bunch
of
different
implications
that
you
can
make
in
order
to
make
the
problem
slightly
easier
and
we
discuss.
We
can
discuss
this
at
length,
but
for
understanding
the
purposes
of
the
algorithm.
B
The
only
thing
I
really
want
you
to
know
is
that
when
we
know
that
a
neutral
surface
should
connect
somewhere
between
somewhere
within
a
layer,
you
just
have
to
find
out
where
this
Delta
row
equals
zero
somewhere
within
that
layer.
B
So
now
we're
getting
to
the
part.
So
now
that
we
have
the
sub
layer,
we
can
actually
calculate
a
flux
in
it,
and
so
this
is
where
so.
This
is
the
equation
that
we
use
to
calculate
that
diffusive
flux
right.
So
we
have
this.
We
have
a
Kappa,
or
so
just
a
diffusivity
times
the
an
effective
thickness
of
the
sub
layer,
which
I'll
explain
in
a
second
just
and
then
just
times
the
diff
times
the
Tracer
gradient
itself
right.
B
So
what
this
actually
looks
like
then,
on
this
bottom
right
hand
figure.
Is
that,
by
the
way
that
we
calculate
these
c
bar
terms
is
by
taking
an
average
over
those
polynomial
reconstructions.
So
remember
that
there
is
a
polynomial
representation
of
what
the
Tracer
concentration
is
in
the
vertical
and
we
average
that
reconstruction
between
the
between
the
two
interfaces
of
the
model.
B
So
that's
shown
on
the
right
hand
the
right
hand
panel
there,
where
you
have
the
c
bar
J,
K,
plus
1
terms
and
the
C
J,
plus
one
k
minus
one
terms,
and
so
that's
how
you
have
that's,
how
we
calculate
the
Tracer,
the
difference
in
the
tracers
there,
this
H
effective.
So
this
is
the
effective
thickness
of
the
sub
layer
to
be
calculated.
By
doing
a
harmonic
mean
of
the.
B
Of
the
sub-layer
in
the
left-hand
column
and
on
the
right
hand,
column.
So
the
only
thing
really
to
really
point
out
on
that
is
that
this
allows
us
to
say
that
if
the
layer
is
really
really
thin,
that
or
if
the
layer
on
one
side
goes
to
zero,
then
there
is
no
diffusive
bucks
right,
so
you
would
never
diffuse
anything
into
a
layer
that
doesn't
have
anything
else,
and
that's
mainly
the
big
point
that
we're
trying
to
make
here.
But
this
is
essentially
the
way
that
we
calculate
the
flux
within
that
sub-layer.
B
One
thing
to
note
in
the
solid
Rhythm
right
is
that
we
is
that
these
sub
layers
can
actually
connect
different
parts
of
the
different
parts
of
the
water
column.
You
know
so
in
this
particular
case
here.
B
The
top
right
layer
is
actually
connected
to
the
bottom
left
layer,
and
you
can
actually
then
also
connect
multiple
layers
to
in
one
column,
to
another
column.
So
this
is
what's
demonstrated
in
this
slide
here,
and
this
is
how
we
then
would
actually
update
the
Tracer
state
within
a
cell.
B
So
in
this,
in
this
particular
example,
here
we
have
a
a
single
layer,
a
single
cell,
on
the
left
hand,
side
that
is
actually
has
connected
via
these
sub-layer,
so
two
different
parts
of
the
water
column
on
the
right
hand,
side,
but
we've
calculated
these
neutral
fluxes
within
the
sub
layer,
right
oops.
So
there's
only
two
fluxes
actually
in
play
here.
So
there's
this
FN
gamma
flux.
So
that's
the
flux
within
the
first
layer
of
sub
layer
and
then
there's
a
second
one,
one
within
the
second
sub
layer.
B
B
Case,
if
you
assume
that
the
arrows
represent
both
magnitude
and
Direction,
you
can
add
this
up,
on
the
left
hand,
column
to
say
that
F
and
Gamma,
plus,
f
n
plus
one
gamma,
actually
equals
zero,
because
these
two
arrows
are
are
equal
length.
So
part
of
this
Tracer
is
leaving
the
cell,
but
it's
also
getting
replenished
from
the
other
sub
layer.
B
However,
on
the
right
hand,
side,
the
top
cell
will
actually
increase
in
transfer,
because
it's
getting
a
net
Flux
Of,
Tracer
n,
due
to
neutral
diffusion
and
the
bottom
cell
will
actually
decrease
in
trace
the
value,
because
it's
having
Tracer
flux
out
of
it.
So
this
is
a
really
important
thing
to
take
away
too
is
that
these
fluxes
can
be
not
are,
can
be
non-local
in
the
vertical.
So,
for
example,
this
sub
layer
can
this.
B
This
k
equals
one
way
or
on
the
left-hand
side,
is
connecting
to
K
minus
1
layer
and
the
K
plus
one
layer
on
the
other
side
on
the
other
side
of
the
column,
and
this
also
demonstrates
that
multiple
sub
layers
can
contribute
flux
of
multiple
layers
and
contribute
fluxes
to
a
single
layer.
Another
column-
and
you
can
actually
in
this
using
this
algorithm,
you
can
actually
diffuse
an
entire
column
into
one
particular
layer
of
the
other
column.
C
B
B
Way
of
conceiving
how
we
calculate
neutral
fluxes,
but
it
also
does
get
around
the
problem
of
the
rotated
diffusion
because
we
never
calculate
a
slope
here,
we're
never
rotating
things,
there's
only
ever
horizontal
fluxes
that
are
coming
into
and
out
of
a
cell
but
to
neutral
diffusion,
and
so
then
I
just
want
to
close
real
quick
to
show
how
this
what
this
actually
looks
like.
So
we
came
up
with
a
couple
different
test
cases
where
we
used
the
the
Tios
10
non-linear
equation
of
State.
B
We
if
this
is
implemented,
mom
six,
where
the
only
physical
process
that's
happening,
is
diffusion
and
we've
discriptanized
the
model
in
two
different
ways:
one
we've
done
the
continuous
isoprignal
and
we
diffuse
along
the
model
surfaces,
and
then
we
also
have
this
set
up
as
a
z-star
coordinate
a
model
where
we
do
then
do
this.
We
apply
this
Mutual
diffusion
algorithm.
So
hypothetically,
if
our
neutral
diffusion
algorithm
is
working
perfectly,
it
should
look
like.
B
We
should
essentially
get
the
same
answer
as
the
continuous
isopypnol
case,
where
we're
doing
diffusion
along
the
layer,
and
so
what
this
looks
like
is
that
this
is
basically
a
very
Clinic
looking
Zone,
where
you've
got
where
the
black
lines
on
the
right
hand,
side,
show
the
isopycles
and
then
the
colored
backgrounds
are
salt
on
the
top
panel
and
temperature
on
the
bottom
right.
So
right,
so
you
have
these
places
where
isocalines
and
isotherms
are
both
intersect,
intersect,
isopycles.
B
So
if
you
run
this
for
it's
a
steady
state
or
pseudo
steady
state,
this
is
essentially
what
you
get
now.
This
is
demonstrating
this
TS
diagram
on
the
left
hand,
side
where
the
black
dots
are
the
original
space
of
the
model,
and
then
the
red
dots
are
the
final
state.
B
So,
as
you
can
see,
when
we
run
the
steady
state
now
you
only
have
a
single
temperature
and
selenium
value
for
any
given
isopicinal
Contour,
you
know,
and
so
instead
of
having
this
envelope
of
TS
points,
you
have
a
single
line
that
you
have
a
single
line.
It's
all
collapsed
onto
a
single
line
and
that's
essentially
what
we
expect
from
diffusion.
B
If
you
will,
if
you
prefer
to
think
about
it,
based
on
the
the
TNS
distribution
spatial
distributions,
that's
shown
on
the
top
right
hand,
side
for
temperature,
where
you
can
see
now
that
those
that
those
black
lines
which
represent
ice
pickles,
they
have
not
moved
in
the
slightest
from
the
beginning,
part
of
from
the
initial
State.
However,
the
temperature
lines,
the
isotherms,
are
now
parallel
to
those
isopycles
right.
So
we
have
this
homogenization
of
both
temperature
and
salinity
on
these
isopycles.
B
Kind
of,
and
that's
exactly
what
we
expect
as
a
result
of
diffusion
along
my
spiritual
diffusion.
Oh.
A
B
Should
make
I
should
make
mention
that
to
compare
the
two
ways
of
doing
it,
we've
actually
set
this
to
be
to
have
breakfast
pressure
in
2000
decibar,
but
they're,
that's
purely
just
to
demonstrate
and
compare
against
the
other,
the
other
case,
so
there
if
we
run
with
neutral
diffusion,
where
we
use
the
average
pressures
between
water
Parcels
to
calculate
the
difference
in
density,
we
get
very
similar
results
and
that's
actually
what
we
end
up
doing
in
the
little
course
resolution
configurations
of
mom
six,
so
just
lastly,
to
close
is
that
we
can,
as
I
said
before,
like
if,
like
epipnal
or
epineutral
diffusion,
should
not
do
any
diet.
B
Picnol
diffusion
diffusion,
so
we
come
up
with
a
way
of
trying
to
diagnose
how
much
this
is
actually
done
by
by
comparing
it
to
a
very,
very
high
resolution,
isopicol
only
case,
and
then
we've
run
this
model
for
both
that
zedstart
coordinate
where
we
calculate
it
where
the
model,
the
model
layers
are
all
isopicinal
or
continuous
isopycnel.
And
then
we
have
a
plot
done
of
this.
In
that
second
case,
where
we
have
a
zed
star,
coordinate
with
constant
spacing
but
apply
the
neutral
diffusion.
B
So
this
is
how
we
calculate
this
first,
this
spurious
diffusivity,
but
but
the
main
takeaway
from
this
bottom
right
hand
line.
Is
that
this
black
curve,
this
black
line,
which
represents
a
long
layer
diffusion
when
the
model
is
configured
in
its
picknel
space,
basically
follows
exactly
on
those
these
blue
dots,
which
is
the
same
X
the
same
Spurs
diffusivity
that
you
get
from
running
our
neutral
diffusion
algorithm
in
a
z,
star,
coordinate.
B
B
B
I
I
probably
should
have
proof
read
that
line
a
little
bit
more,
but
essentially,
we've
demonstrated
this
an
idealized
test
case.
So
the
only
the
main
takeaway
home
from
this
is
that,
instead
of
doing
a
rotated
upward
approach
communication,
we
construct
sub
layers
of
the
model
where
we
think
that
that
are
bounded
by
by
surfaces
of
neutral
density.
B
And
then
we
calculate
fluxes
along
that
and
we
come
up
with
a
scheme
that
still
realizes
the
the
impact
of
eddies
on
on
diffusing
tracers,
but
still
maintains
all
these
things
that
that
include,
monitor
that
it's
with
monotonously
that
don't
introduce
any
new
extrema
and
is
still
relatively
cheap,
though
it
is
slightly
more
expensive
than
the
rotated
operator
approach,
and
with
that
I
am
happy
to
take
any
questions
or
field
any
clarifications
that
people
might
want.
A
Yeah
Andrew,
thank
you.
This
is
yokan
I
had
a
quick
question
just
for
clarification
purpose
when
you
said
that
there
is
no
slope
limit
or
any
kind
of
tapering
with
this
approach,
if
you're,
essentially,
your
salting
algorithm
can
be
such
that
you
may
be
essentially
diffusing
from
the
very
bottom
of
one
particular
column
to
the
very
top
of
the
next
neighboring
column.
Yes,.
C
A
Right,
so,
given
that
your
scheme
is
explicit
in
physical
terms,
when
you
take
a
water
blob
from
the
very
bottom
and
all
the
way
put
it
up
there
up
next
to
a
column,
do
you
have,
you
may
be
essentially
physically
taking
that
water
parcel
faster
than
that
would
be
allowed
by
the
top
step
of
the
scheme?
Do
you
have
to
worry
about
in
that
case,
or
how
does
the
skin
take
care
of
it,
given
that
it's
an
explicit
scheme.
B
So
we
we
don't
limit
the
kind
of
that
rate
that
it
can
happen.
The
only
thing
that
we
end
up
doing
is
that
we
do
set
a
CFL
limit
on
the
diffusive
part
of
it:
okay,
okay,
so
to
essentially
make
sure
that
we
don't
completely
deplete
a
cell
and
turn
it
negative.
Okay,.
B
Yeah
yeah,
so
so,
essentially,
what
it
boils
down
to
is
that
you
can't
it's
it's
chosen
in
such
a
way
that
the
Tracer
concentration
can't
go
negative,
but
there's
no.
If
you
try
to
turn
it
into
like
an
equivalent
effective
flux.
There
is
no
limit
on
that.
So,
hypothetically
yeah,
you
could
you
could
you
know
very
rapidly,
move
from
over
the
course
of
five
thousand
meters.
If
your
time
step
was
like
30
seconds,
you
would
still
have
a
flex
associated
with
diffusion,
but
yeah.
C
C
So
Andrew
was
hit
the
hit
the
nail
on
the
head.
The
key
thing,
though,
is
that
the
CFL
constraint
is
based
entirely
on
the
horizontal
grid
spacing
and
the
time
step,
and
this
diffusivity
Kappa
out
front.
It
doesn't
have
anything
to
do
with
the
layer
structure.
That's
taken
care
of
by
this
effective
thickness
over
here
on
the
right
hand,
side
which
is
well
there's
a
I
guess,
there's
a
factor
of
two
missing,
but
this
is
the
harmonic
mean
that
that
takes
care
of
the
cases
where
you
have
thin
layers
interacting
with
thick
layers.