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From YouTube: Stephen Griffies MOM6 Webinar 3 13 20
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A
There
all
right
so
this
is
this-
is
a
version
of
a
talk
I
gave
at
Ocean
Sciences,
as
well
as
in
Hamburg
and
in
Grenoble.
So
if
any
of
y'all
were
there,
there'll
be
some
repeats,
and
this
is
from
Bob,
allister
and
I,
and
a
lot
of
it
is
taken
from
a
manuscript
that
is
in
review
at
James.
Talking
about
the
method
of
how
we
solve
the
equations
in
mom
6.
A
Thinking
of
it
more
of
a
fluid
mechanic
perspective
than
a
numerical
perspective,
so
I
hope
that
I
can
convey
some
of
the
sort
of
higher
level
conceptual
understanding
of.
What's
going
on
with
Mom
six
to
those
that
may
want
to
understand
that
and
not
really
able
to
dive
into
the
numerical
details,
kind
of
which
is
what
I
found
myself
in
a
few
years
ago.
Next
slide.
A
So
there's
some
key
points
to
the
talk
and
I
want
to
remind
people
of
some
of
the
motivation
for
doing
what
we
do
in
mom
6
and
one
of
the
big
motivations
stems
back
from
a
20
year
old
problem,
namely
experience
diet,
Picto
mixing
that
can
be
exacerbated
in
Zed,
like
vertical
coordinate
models
or
Sigma,
coordinate
models
or
models
that
don't
have
sort
of
a
quasi-isopicinal
nature
to
it,
and
the
dynamical
core
formulation
that
I'm
going
to
talk
about
is
what
we
call
vertical
lagrangian
remapping
and
a
lot
of
people
call
it
Al
ale
for
arbitrary
lagrangian
orlarian
and
like
to
make
a
distinction
here
and
I'll.
A
Try
to
highlight
that
as
we
go
along
and
I'm
going
to
offer
conjecture-
and
it's
nothing
proven
in
this.
This
work.
But
it
is
something
that
we're
hoping
is
going
to
be
proven
in
the
community
over
the
next
decade
is
that
the
vertical
lagrangea
remapping
provides
an
appropriate
hybrid,
vertical
coordinate
framework
to
perhaps
even
optimal
framework,
to
solve
the
spurious
mixing
problem
in
a
way
that
allows
us
to
also
solve
the
global
ocean
circulation
problem.
A
Numerically,
you
can
solve
this
previous
mixing
problem
for
the
most
part
by
had
gone
100
isopygnol,
but
you
compromise
your
ability
to
to
do
the
coastal
problem
in
the
high
latitude
problem
when
you
go
isopyctal
100
percent,
whereas
a
hybrid
approach,
that's
implemented
through
vertical
lagrangea
remapping,
we
think
is
a
good
framework
to
do
to
do
both.
Okay
next
slide
next
slide.
A
Next
slide:
okay,
I'm
going
pretty
fast
here
anyway,
the
the
mixing
problem
now
this
is
this-
is
rudiments
for
many
people,
but
I'm,
hoping
that
it
will
be
new
to
some
so
we're,
basically
in
a
point
in
the
community
that
we
are
able
to
make
physical
statements
about
mixing
in
the
ocean.
A
This
is
contrast
to
maybe
20
years
ago
where
there
was
really
lots
of
unknowns
about
what
were
the
mechanism
for
mixing
I
can't
say
that
we
100
understand
everything,
but
this
picture
here
from
a
review
article
from
Jennifer
McKinnon,
which
is
the
result
of
a
climate
process
team
that
finished
a
few
years
ago,
is
exemplary
of
a
lot
of
the
physical
processes
that
we
have
a
grip
on
now,
and
we
have
some
rudiment
parametrizations
of
these
processes.
A
There's
more
work,
that's
going
on
with
a
big
center
of
excellence
in
Germany,
also
looking
at
similar
questions,
and
we
really
are
at
the
point
where
the
physics
is
going
to
be
compromised
in
a
way.
That's
not
good.
If
we
have
spurious
mixing
coming
from
the
numerics
that
that
can
sometimes
swamp
the
physics.
So
we
need
to
be
mindful
of
spurious
mixing
next
slide
so
just
to
frame
the
problem
you
have
and
it's
coming
through
advection.
That's
what
I'm
talking
about
here.
A
A
So
it's
a
key
concern
for
climate
in
particular,
because
this
spurious
mixing
can
accumulate
over
long
periods
of
time
and
lead
to
degradation
of
your
thermocline.
For
example,
you
can
degrade
your
Tracer
concentration
in
a
way.
That's
just
not
physically
desirable
and
it's
reduced
when
you
use
a
quasi-isopycno
vertical
coordinate,
because
air
stop
at
the
interface
actually
using
really
yeah.
But
again,
as
I
said
earlier,
an
isopytonal
approach,
100
isopytonal,
is,
is
difficult,
if
not
impossible
to
do
the
world
ocean
circulation,
so
we're
looking
for
something
different
next
slide
foreign.
A
So,
let's
this
slide
I'm
going
to
skip
through
a
couple
of
the
slides,
really
quick,
there's
methods
to
diagnosis.
Various
mixing
going
back
to
this
paper,
I
wrote
with
Bob
and
Ron
pakanowski
20
years
ago,
and
just
diagnose
the
magnitude
of
the
mixines
that
we're
getting
with
certain
operators
and
it
can
be
upwards
of
10
times
larger,
maybe
even
larger,
in
some
cases
10
times
larger
than
what
you
would
want
from
a
physical
sort
of
background
10
to
the
minus
5
meters
square
per
second
mixing
level.
A
So
what
we
have
found
in
some
of
this,
the
studies
that
came
subsequent
to
that
in
particular
a
paper
by
illichak
at
all
2012,
is
that
there
is
a
real
important
need
to
combine
your
grid
Reynolds
number,
namely
the
friction
you
use
or
the
viscosity
you
use
in
a
model
and
the
spurious
mixing.
A
That
scheme
is
going
to
introduce
mixing
to
smooth
out
the
obvection
operator
to
keep
you
from
producing
extrema.
And,
of
course,
if
you
have
a
an
operator,
that's
that's
linear,
that's
not
flux,
corrected
or
flux.
Limited
then
you're
going
to
get
a
lot
of
dispersion
error
that
creates
negative
and
positive
mixing,
quite
like
the
old
days
of
second
order,
Center
difference
of
action
where
you
get
huge
extrema
of
water
masses.
So
there
is
a
real
need
in
any
of
these
simulations
to
be
mindful
of
the
grid.
Reynolds
number
you
can't
there's
no
free
lunch
effectively.
A
You
can't
just
run
with
reduced
numerical
friction,
even
though
the
model
is
stable.
It's
not
blowing
up.
It
still
may
be
producing
lots
of
grid
noise.
Okay,
so
next
slide.
A
A
So
just
the
general
points,
I'm,
sorry
previous
slide,
spurious
mixing
diagnosed
insperous
mixing
is
useful
for
understanding
its
character,
but
it
is
insufficient
to
remove
the
problem
higher
order
numerics
help,
namely
as
a
paper
by
hillidal
in
2012,
where
they
looked
at
these
the
pray,
their
advection
scheme,
which
they
claim
they
solve
the
spurious
mixing
problem
with
using
the
prey,
the
reduction
scheme,
and
they
did
perhaps
solve
it
in
a
flat
bottom
numerical
model
without
any
real
yeah,
without
realistic,
forcing
going
on
with
a
linear
equation
of
state.
A
But
you
still
need,
when
you
add
passive
tracers
like
biology.
It's.
We
have
found
that
you
need
to
have
flux
limiters
and
when
you
add
flux
limiters
to
the
trailer
scheme,
it
reduces
the
accuracy
of
that
scheme
down
to
something
like
a
piecewise,
parabolic
method,
limited
piecewise,
parabolic
method.
So
there's
not
that's,
not
the
the
solution
that
we
have
found.
A
Really.
You
have
to
maintain
a
modest
grid,
Reynolds
number
and
use
a
good
adduction
scheme.
In
addition,
and
the
pre
in
the
the
problem
arises
from
eviction
both
in
the
vertical
and
the
horizontal
there's
a
paper
by
Gibson
at
all
that
decomposed
in
mom
6
the
contribution
to
spurious
mixing
from
the
vertical
remapping,
as
opposed
to
the
hor,
in
contrast
to
the
horizontal
advection
along
layers.
A
So
anyway,
Let's
let's
skip
to
the
the
next
slide,
because
we
that's
more
more
motivation.
Just
to
summarize
the
state
of
the
science
of
the
spurious
mixing.
So
we'll
talk
about
the
next
slide,
this,
the
vertical
LaGrange
and
remapping.
A
So
we're
going
to
focus
on
the
vertical
and
that's
really
been
the
theme
of
a
lot
of
the
work
that
gftl
has
done.
The
last
really
since
Hallberg
came
to
gftl
in
the
90s,
really
focusing
on
the
vertical
and
the
physics
as
well
and
and
isopic
models
have
their
role
in
the
climate
problem.
But
again
we
have
found
that
there
are
limitations
to
that.
A
We
built
a
climate
model
with
with
the
gold
isopy
model
and
it
was
a
very
respectable
model,
but
they
have
their
Prime
their
problems
in
high
latitude
on
stratified
regions
which
argue
for
hybrid
and
the
vertical
or
Larry
the
arbitrary,
the
vertical,
arbitrary
or
lagrangian
eulerian
method.
The
ale
method,
which
includes
a
special
case
of
vertical
lagration
remapping,
is
the
strategy
we've
taken.
A
So
let's
propose
a
terminology
and
that's
what
we
use
in
our
paper.
That's
being
reviewed
right
now.
Quasi-Arlarian
is
something
like
what
we
had
in
what
we
have
in
mitgcm
mom,
5
ROMs,
where
you
solve
the
equations
very
similar
to
what
almost
like
what
Kirk
Bryan
did
in
the
late
60s.
But
you
allow
for
the
vertical
coordinate
to
go
up
and
down,
but
there
are
larion,
it's
an
eulerian
method,
vertical
arbitrary,
lagrangian
or
larion.
Is
you
can
Implement
that,
with
or
without
remapping?
A
You're
following
particles
at
some
point,
you've
got
to
renormalize
or
re-initialize
your
coordinate
in
order
to
maintain
the
Integrity
of
your
numerical
simulation,
that's
done
by
a
remapping
in
the
ocean.
We
do
this
in
the
vertical
only,
whereas
in
some
communities
like
the
bomb
simulators
at
Los
Alamos,
they
do
a
full
3d
or
larion
lagrangian
approach,
LaGrange,
an
alerting,
poetry,
full
3d,
lagrangian
approach,
and
so
they
have
a
much
more
complicated
re-gridding
remapping
problem
than
we
do.
A
A
There's
a
velocity
V
of
the
fluid
itself
and
a
velocity
V
super
B
of
the
boundary
of
that
region,
and
you
take
the
relative
velocity
of
the
two
to
give
you
the
amount
of
advection
through
that
domain.
Okay
and
the
momentum
equation
has
a
similar
expression
where
you
have
body
forces
and
contact
forces.
A
It
could
be
a
fixed
cell,
like
we
did
in
the
old
days
with
top
and
bottom
and
sides
all
fixed
in
time
like
we
did
in
the
Zed
models,
we
can
have
the
quasi-oralarian
models
where
the
top
and
bottom
can
go
up
and
down
on
a
special
specified
way,
or
we
can
have
different
more
General
ways
of
doing
it,
like
we
have
in
the
lagrangian
approaches
models,
typically
formulate
the
scalar
prognostic
budgets
for
extensive
quantities
like
heat
and
salt,
not
temperature
and
salinity.
Okay.
A
So
you
may
see
me
in
other
contexts
being
a
little
bit
anal
about
this
distinction
and
I
think
it's
important
to
remember
that
we're
solving
a
heating
equation
in
an
assault
equation,
not
a
temperature
equation
and
a
salinity
equation
that
makes
all
the
difference
in
the
world.
What
makes
a
big
difference
when
you're
talking
about
diagnostic,
diagnosing
budgets
and
a
lot
of
what
I
do
now
is.
Is
analysis
of
these
simulations
and
talking
about
budget
analysis
is
a
big
deal.
A
However,
when
we're
doing
the
velocity
or
the
momentum,
we
have.
We
often
reformulate
things
in
terms
of
vector
and
variant
velocity
equations
for
various
reasons,
such
as
Southern
energy
entropy,
is
a
favorite
numerical
method
that
some
people,
like
so
I'm,
not
going
to
talk
about
the
momentum
equation
or
the
velocity
equation
in
this
presentation.
But
it
is
a
distinction
that
is
important
to
keep
in
mind
if
you're
talking
about
diagnosis,
diagnosing
those
terms
so
advection
again
refers
to
the
transfer
of
the
fluid
relative
to
the
grid.
A
A
fully
lagrangian
method
has
zero
advection,
okay,
but
again
follow
the
LaGrange
and
at
some
point
it's
going
to
fall
flat
on
its
on
its
face,
because
the
the
grid
is
going
to
evolve
so
far
that
you're
losing
your
ability
to
simulate
the
flow.
So
you
have
a
turbulent
flow
or
you
just
have
an
unstratified
region.
A
So
this
is
one
of
my
favorite
figures,
which
I
think
a
lot
of
people
said
is
a
is
they're
into
is
a
good
intuition
for
the
arbitrary
LaGrange
and
Larry
and
method.
So
it's
a
little
more
specialized
than
the
previous
kidney
shaped
figure
that
we
had
and
this
one
now
we
have
three
figures
left
middle
and
right,
the
left
figure.
You
say
you
have
a
square
region
of
fluid
okay
and
now,
let's
put
into
that
region
of
fluid
a
grid
line
and
let's
paint
a
blue
and
a
yellow
stripe
in
that
grid.
A
Okay,
so
that's
the
initial
fluid
region
and
the
initial
grid
all
right.
Now,
let's
let
that
fluid
evolve
over
some
small
but
non-zero
time
difference.
Okay,
as
that
fluid
evolves,
the
grid
may
or
may
not
evolve
with
the
fluid.
In
this
particular
example,
the
grid
is
evolving,
with
the
fluid
and
the
fluid
itself.
The
yellow
and
the
blue
stripes
are
evolving
and,
as
I've
drawn
this
here,
they're
evolving
together,
okay,
but
it
need
not.
They
need
not
evolve
together.
A
A
Okay,
if
your
grid
evolves
in
a
lagrangian
way,
as
I've
said
a
few
times,
you
eventually
have
to
re-grid,
okay
and,
as
you
re-grid,
your
Grid
in
this
case
moves
back
to
some
re-initialized
position,
and
you
see
that
the
state
of
the
fluid
is
distinct
now
from
the
grid.
All
right,
and
so,
however,
your
fluid
needs
to
be
known
needs
to
be
placed
back
on
the
grid
you
need
to
have
that
grid
has
to
carry
information
about
the
fluid,
and
so
that's
the
remap
step.
The
remap
step
is
not
moving
the
fluid
itself.
A
It's
just
moving
where
you
know
the
fluid
exists.
The
points
where
you
have
the
information
about
the
fluid
okay,
so
I,
just
let's
see
if
I'm
going
to
read
here
if
there's
anything
new
step
one
if
the
grid
moves
or
the
flow
it's
lagrangian
step.
Non-Lagrange
and
grid
motion
is
also
considered
by
certain
ale
approaches,
such
as
what
we
have
an
impasse
in
Nemo
step:
two,
the
re-grid
and
remap
step.
They
come
together,
they
idealize.
Ideally
they
do
not
alter
the
ocean
state.
A
But,
of
course,
due
to
numerical
truncation,
you
will
have
some
altering
of
that
state.
The
remap
step
operationally
equals
to
advection
it's
a
transport
relative
to
the
grid.
Remember
that
sort
of
Mantra
that
I
mentioned
earlier.
What
advection
is
it's
transport
relative
to
a
grid?
A
A
So
we
distinguished
this
the
solution
methods,
here's
a
here's,
a
typical
grid
cell
and
an
ocean
model.
That's
that's
sitting
away
from
the
boundaries,
so
we
have
I'm
using
the
symbol
Sigma
for
the
or
for
the
coordinate
grids,
and
we
have
outward
normals
and
X.
Y
and
Z
is
oriented
as
shown
if
the
lateral
boundaries
are
and
typically
the
lateral
boundaries
in
Ocean
Models
are
fixed.
A
We
don't
have
fully
moving
grids
in
most
Ocean
Models,
at
least
the
ones
that
we're
talking
about
here
and
so
the
lateral
boundaries
where
flow
moves
relative
to
that
fixed
boundary.
That's
horizontal
advection,
just
like
familiar
horizontal
advection,
rigid
lid
Zed
models
have
the
vertical
fixed
and
they're,
also
flat,
well
effectively
flat,
and
so
any
vertical
motion
is
a
vertical
advection
in
these
models.
Likewise,
for
free
surface
models,
vertical
objection
is
through
the
interior
vertical
cell
faces,
and
this
the
top
of
the
ocean
is
the
one
that
moves
we
have.
A
Analytically
specified
coordinates
such
as
Z,
star
or
Sigma
Sigma,
coordinate
models
where
you
you
effectively
filter
out
the
barotropic
Motions
in
these
models.
Isopy
no
layers
are
determined
by
layers
that
are
follow.
Interior
layers
that
follow
the
the
fluid
itself
and
more
General
ale
allows
for
the
the
top
and
bottom
interfaces
to
be
arbitrary,
so
this
is
kind
of
a
categorization
of
the
different
methods
based
on
just
looking
at
how
we
specify
the
the
grid
motion
itself,
the
grid
being
the
top
and
bottom
interface
of
this
cell
in
an
ocean
model.
A
Let's
skip
this
slide
so
now
we
specialize
a
little
bit
more
to
show
a
typical
flavor
of
a
vertical
lagrangea
remap
method,
and
this
is
a
depiction
of
a
vertical
and
horizontal
layering
of
the
ocean
according
to
a
grid,
and
we
have
again
initial
State
and
various
subsequent
States.
So
let's
talk
through
this
figure
here.
This
is
also
a
very
hopefully
intuitive
figure
for
what's
going
on
in
mom,
6
and
high
com.
A
So
the
next
line
is
an
evolution
of
that
Ocean
State
according
to
say,
a
wave
that
goes
through
this.
The
fluid
itself
say
it's
an
internal
gravity
wave
that
moves
the
interfaces
of
the
isopycles
up
and
down,
and
this
first
step
of
the
algorithm.
The
grid
follows
the
motion
of
the
fluid
okay.
That's
we've
already
talked
through
this,
but
this
is
now
getting
more
specific
into
the
mom
vertical
isopygnal
approach,
vertical
lagrangian
approach,
the
next
step
we
re-grid
okay.
A
Now
we
don't
have
to
re-grid
every
time
step,
but
in
this
particular
example,
let's
just
assume
this
is
the
regrad
step,
all
right,
so
we're
redefining
the
grid.
The
dashed
lines
now
are
the
ocean
state
and
the
solid
line
is
the
grid,
the
new
grid
and
now
the
third.
The
fourth
panel
shows
a
remapping
of
the
state
onto
the
horizontal
lines.
A
Okay,
we
still
see
the
dashed
lines
for
the
the
previous
grid,
but,
as
we
see
I,
don't
know,
if
you
can
tell
you
can
probably
tell
going
to
the
final
state
where
we
remove
all
the
the
old
grid
lines,
you
can
see
that
there's
a
bit
of
diffusion
in
the
process
of
that
regrit,
okay
and
so
we're,
because
we
have
a
finite
vertical
representation
find
that
vertical
grid,
and
so
there
is
some
diffusion
in
that
remapping
step
all
right
and,
of
course,
a
lot
of
people
ask
well
why.
A
A
If
you
had
fluid
mechanics
where
it
was
just
pure
isopytonal
fluid
flow,
then
your
vertical
lagrangian
model
would
just
you
would
choose
isopycles
as
your
vertical
coordinate
instead
of
a
z
coordinate
as
we
have
here,
but
this
is
a
just
the
example
that
I
wanted
to
show
that
allows
for
you
to
see
the
the
different
steps
in
the
vertical
lagrangian
approach
and
hopefully
gives
you
a
sense
for
why
there
is
some
mixing
associated
with
that
remap
step.
A
A
Regrad
remap
step,
which
is
you
know,
require
that
re-grid
remap
step
requires
a
lot
of
technology
and
it's
an
important
step
to
get
the
numerix
right
and
it's
something
that
evolving
a
model
such
as
impasse
and
Nemo
from
Zed
like
models
to
something
more
General.
It's
a
totally
rational
approach
to
say:
well,
let's
do
some
something
in
between
the
folio,
LaGrange
and
remap
next
slide.
A
So
again,
where
is
advection?
Where
is
the
the
the
invection
across
coordinate
surface?
That
is
again
in
the
remap
step?
It's
the
movement
of
your
representation
of
the
grid
of
a
state
relative
to
the
grid.
Okay,
the
remap
step.
You
are
changing.
How
you're,
representing
your
your
state
from
one
grid
to
another
and
so
you're,
moving
through
a
grid
line,
the
fluid
the
flow
is
moving
through
grid
line,
even
though
the
state
itself
ideally
is
not
changing.
It's
the
representation
of
that
state
that
changes
and
so
to
diagnose.
Here's.
A
Here's
putting
my
analysis
hat
on
if
I
want
to
diagnose
the
full
advection
operator
acting
on
a
tracer
I
would
have
to
include
that
re-grid
that
remap
step
okay
and
it's
written
here
in
equation,
11
as
just
remapping.
It's
a
word.
You
can't
really
write
down
a
you
know
a
clean
operator
form
of
it
other
than
just
putting
something
like
vertical
advection,
but
it
is
it's
it's
in
quotes.
Okay,
sorry
I've
got
some
slack
people
making
noise
here
and,
and
likewise
it's
interesting
to
note
that
there
is
no
CFL
associated
with
vertical
remapping.
A
Okay,
so
that's
very
useful
for
fine
vertical
grid
spacing.
But,
of
course
remember,
stability
does
not
imply
accuracy,
but
even
having
said
that,
that
is
really
the
the
piece
of
technology
that
allows
us.
A
If
we
write
our
numerics
allowing
for
remapping
over
one
grid
grid
cell
in
the
vertical
allows
which
we
have
in
mom
six,
it
allows
us
to
have
Vanishing
layers
allows
us
to
have
ice
shelves
that
that
move,
the
grounding
line
interactively
every
time
step
and
it's
a
pretty
powerful
thing
moving,
be
you
know,
biting
the
bullet
and
going
lagrangian
in
the
vertical
allows
you
to
have
some
real
power
beyond
just
the
fully
eulerian
approach
or
the
Quasi
orlarian
or
the
the
non-remapping
approach
you
can.
A
Having
said
that,
you
can
do
the
the
vet,
you
can
do
the
the
vanishing
layer
problem
effectively
in
the
other
approaches,
but
coming
at
this
from
an
isopytonal
approach,
where
you
always
had
to
solve
the
vanishing
layer
problem,
gives
us
a
little
step
up
on
that
problem.
So
the
coastal
Estuary
problem,
where
you
have
wetting
and
drying,
is
something
that
is
well
handled
in
the
mom6
context,
because
we
do
that
the
numerix
built
in
the
vanishing
layers
from
the
beginning.
A
So
I've
gone
for
about
25
minutes.
I.
Think
I'd
like
to
to
to
spend
a
couple
of
minutes
on
just
showing
a
few
numerical
equations
here
and
I'm
going
to
talk
through
this
pretty
quickly
and
if
there's
any
questions
that
you
can
come
back,
if
you
need
to
follow
up
on
anything
the
way
that
we
can
sort
of
characterize
from
a
numerical
perspective,
the
different
algorithms,
it's
just
by
looking
at
the
thickness
equation
and
the
thickness
way
to
trace
the
equation.
A
So
if
we
get
the
Dynamics
allow
just
allow
the
Dynamics
to
be
given
to
us
from
somewhere,
okay-
and
there
are
three
methods
that
the
community
has
developed
and
we
have
this
quasi-arlarian,
which
is
what
we
do
in
mitgcm,
mom5
Nemo
and
then
the
vertical
AO
without
remapping
is
Nemo.
Zed,
tilde
and
impasse
o
and
the
vertical
LaGrange
and
remapping
is
high
common
mom
six.
A
So
here
is
the
thickness
equation
and
the
thickness
weighted
Tracer
and
the
W
Sigma
dot
is
what
we
call
the
dios
surface
velocity
it's.
The
vertical
velocity
in
the
sigma
coordinates
the
generalized
vertical
coordinates
and
the
first
term
on
the
right
hand,
sides
of
both
equation.
12A
and
12b
is
the
horizontal
advection,
the
convergence
of
the
advection
within
layer
within
this
generalized
vertical
layer,
Sigma,
okay,
so
next
picture
next
slide.
A
Because
of
time,
let
me
just
really
be
quick
on
this,
so
the
or
quasi-arlarian
approach
you
have
a
vertical
grid
change
due
to
step
in
Step
number
one-
and
in
this
case
it's
just
due
to
say
that
the
evolution
of
the
free
surface,
okay,
like
we
have
in
a
z,
star
or
a
sigma,
coordinate,
terrain
falling
coordinate
model,
and
then
we
calculate
we
diagnose
the
diet,
surface
transformation
just
by
the
continuity
equation,
and
then
we
update
the
horizontal.
A
A
So
next
slide
the
impasse
o
and
Nemo
Z
tilde
approach
is
identical
to
the
previous,
but
the
only
thing
that
changes
is
the
specification
of
the
grid
and
in
this
context,
there's
a
Target
grid
and
a
that
gives
you
this
evolution
of
the
Grid
in
equation
one.
So
it's
a
more
General
layer
motion
that
allows
for
to
be
less
constrained
in
your
vertical
coordinate.
Choice
next
slide.
A
So
the
vertical
lagrangian
eulerian
is
slightly
different,
although
this
is
it
we're
trying
to
write
the
algorithms
in
a
way.
Numerically,
that
corresponds
very
closely
to
the
previous
two.
The
first
step
is
evolving
the
grid,
according
to
just
the
horizontal
evolution
of
the
horizontal
convergence
of
the
thickness
weighted
velocity.
So
it's
layer
of
motion
by
horizontal
advection
and
then
there's
a
calculation
of
the
DIA
surface
transport
and
it's
zero
by
definition
in
the
first
step.
Okay,
so
there's
no
transfer
across
the
layer,
because
it's
a
fully
lagrangian
step
again.
Remember
the
Mantra.
A
The
advection
is
motion
relative
to
a
grid.
If
the
first
part
of
the
algorithm
is
lagrangian,
there
is
no
advection
in
the
vertical
okay,
of
course,
there's
horizontal
advection,
and
so
we
update
the
thickness
based
on
a
purely
adiabatic,
horizontal
advection
step.
The
horizontal
Vector
thickness
update
that's
step
number
three,
and
likewise
we
update
the
thickness
way
to
trace
a
concentration
within
layer
according
to
that
horizontal
convergence
of
thickness
weighted,
Tracer
concentration
and
then
step
number
five.
A
We
have
some
specified
Target
grid
it
in
the
example
that
I
showed,
that
was
our
isopygnol
I
mean
I'm,
sorry
that
was
our
Z
coordinate
our
geopotential,
and
from
that
we
can
diagnose
the
motion
of
the
grid
itself
relative
to
the
fluid
the
W
Sigma
Dot,
and
that's
the
target
minus
the
H
dagger
divided
by
Delta
time,
and
we
can
then
use
that
to
re-grid
the
Tracer
and
that's
the
vertical
lagrangian
approach
in
mom
6.
In
a
nutshell:
I
have
no
dipignal
stuff
going
on
here.
I
don't
have
any
diffusion.
A
The
diffusion
in
fact,
is
added
in
the
step.
If
you
had
Tracer
diffusion,
that's
added
in
Step
number,
four.
Okay,
so
we
have
everything
going
on
in
Step
number
four
in
the
trace
equation,
except
W,
Sigma
dot
is
zero.
Okay,
so
don't
confuse
lagrangian
for
adiabatic.
We
still
have
diopignol
velocities.
We
still
have
dipignals.
We
just
still
have
diffusion
going
on
in
Step
number:
four:
okay,
we
just
don't
have
the
W
Sigma
dot,
w
Sigma
dot
is
zero.
It's
a
grid
motion
is
zero
in
that
lagrangian
step
vertical
grid
motion
is
zero.
A
Another
point
that
I
want
to
make
is
about
jet
McWilliams.
Where
does
that's
that
live
in
this
formulation,
in
particular
the
vertical
piece
of
jet
mcquilliams
or
any
advection
whatever?
It
is
either
Fox
Kemper
at
all
or
jet
McWilliams
the
scalar
equation
without
remapping-
and
this
is
a
sort
of
fish-like,
generalized
or
a
coordinate
layer.
Here,
it's
not
kidney
bean.
This
is
fish
bean.
A
So
we've
got
just
remember.
You
have
a
you,
you
star,
V,
Star
and
W
star,
where
the
star
is
your
Eddie
induced
velocity,
and,
as
you
see
here,
we
add
that
W
star
to
the
DIA
surface
transport
in
a
model
without
remapping.
Okay,
so
ignore
the
equations
below.
But
just
briefly
it's
the
full,
like
we
do
in
Zed.
Coordinate
models
now
go
to
the
next
slide
in
the
layered
approach.
In
the
vertical
LaGrange
and
remapping
approach,
the
only
we
don't,
we
don't
do
GM
as
GM
95.
A
We
do
GM
as
a
layered
model.
With
do
GM.
We
have.
We
interpret
the
the
parametrized
eddy-induced
velocity
as
a
bolus
velocity,
it's
just
horizontal
advection
within
layers.
Okay,
so
it's
a!
U
dagger
equals
U,
plus?
U
bolus!
For
that.
U
dagger
I
should
have
probably
used
a
different
symbol.
That's
the
residual
mean
velocity
and
the
transport
across
the
layer
is
just
due
to
the
the
layer
motion
itself.
A
W
Sigma,
Dot,
okay,
so
that
that
is
an
important
realization
when
you're
thinking
about
implementing
generic
ways
which
we've
done
in
mom
six
and
but
it
is
something
that's
distinct
with
thinking
about
it.
In
a
layered
model
like
an
isopygma
layered
approach,
not
a
zed,
zed,
more
model
approach,
I
think!
That's
it.
If
we
keep
going
I
think
that
sort
of
closing
comments,
so
this
is
from
alistair's
at
all
paper,
and
many
of
you
have
seen
this.
A
This
is
more
a
importance
of
hybrid
or
the
the
potential
importance
of
hybrid,
at
least
insofar
as
what
we've
done
in
our
quarter
degree.
No
Man's
Land
climate
model,
no
man's
land
in
a
sense
of
core
degree,
is
an
awkward
resolution
to
be
running
climate
models,
but
we
think
that
it's
it's
it's
a
useful
resolution.
For
us.
A
I
think
that's!
So
the
prop
the
the
the
solution
that
we've
we
have
in
mom
sticks
for
vertical
coordinates
is
very
general.
The
I
think
that
the
Cutting
Edge
research
numerically
is
in
finding
the
best
vertical
coordinates.
There's
still,
we
don't
claim
that
we
have
the
best
vertical
coordinate
with
the
high
com
like
vertical
coordinate
with
the
Z
Star
going
to
isopycnyl.
A
B
Thanks
Steve
I
think
we
can
take
a
few
minutes
for
questions
here,
see
if
I
can
make
this
work
and
I'm,
maybe
if
Todd
or
Elizabeth
I'm
not
apparently.
A
Yeah,
so
if
you
want
to
diagnose
like
geopotential,
Vertical
Velocity,
is
that
the
question
yeah
yeah,
so
we
thought
we
had
it
a
few
years
ago
when,
when
when
Andrew
Xiao
was
at
jftl,
we
said.
Oh,
we
can
just
do
it
offline.
We
can
do
it
in
Python.
A
We
can
make
everything
sort
of
just
take
the
convergence
of
the
horizontal
transport,
but
we
forgot
that
the
there's
a
Time
derivative
of
the
coordinate
movement
itself,
I
think
you
need
to
probably
do
it
online
and
it's
a
it's
a
piece
of
code
that
we're
hoping
that
you'll
produce.
A
It's
it's
it's
something
that
needs
to
be
done
for
certain
diagnostic
purposes.
Say
you
want
to
if
you,
if
you
want
to
continue
doing
lagrangian
particle
trajectories
with
UV
and
W,
for
example,
you'll
need
to
have
W
so
I
I.
We
know
how
not
to
do
it.
It's
just
a
matter
and
I
think
we
know
how
to
do
it.
It's
just
a
matter
of
getting
the
code
in
together
and
doing
it.
B
D
B
B
Okay:
okay,
anyone
else
any.
C
A
How'd
you
decide
when
and
how
frequently
to
do
the
vertical
remapping
yeah,
it's
a
good
question.
I
think
we
Alistair,
hopefully
you
can
correct
me
I
think
we
do
it
every
four
time
steps.
Is
that
right?
Alistar?
D
It's
four
Dynamics
time
steps
once
we're
doing
well
it
variously
on
the
configuration
resolution,
of
course,
but
we're
currently
doing
an
every
Tracer
time,
step
yeah,
which
is
generally
four
Dynamics
time
steps
or
more.
A
And
so
the
the
decision
to
do
it,
that
way
is
I,
guess
based
on
empirical
and
you
try
to
reduce
the
amount
of
remapping
to
the
to
the
minimal
amount
that
you
can
do
it
is
there
anything
more
objective
than
that
Aleister
yeah,
so
he's
he's
moving
away.
I
I
had
the
same
question
asked
at
me:
Alistair.
Is
there
anything
objective
with
the
the
number
of
regret,
regrading
time
steps
remapping
time
steps
so.
D
I
think
there's
an
accuracy
issue,
you.
Obviously,
if
you
don't
remap
regularly,
you're
distorting
the
the
resolution
as
you
as
you
as
you
move
forward,
I
think
it's
one
of
the
things
where
there's
some
empiricism
involved.
It
would
be
useful
for
someone
to
one
day
do
a
systematic
study
of
of
what
the
best
ratio
is
for
particular
resolutions
and
processes.
It's.
A
Yeah
so
again,
more
turbine
at
the
flow.
You
need
to
be
sure
that
you're
remapping
enough,
but
of
course,
your
your
time,
steps
itself
will
be
smaller
too.
So
it's
an
interesting
sort
of
numerical
research
question
to
determine
that
optimal.
If
there
is
an
optimal.
E
A
It's
it's
a
continuous,
it's
a
continuous,
it's
a
continuous
vertical
coordinate,
just
like
high
com,
and
so
we
have
options
for
kpp.
We
have
okay,
energy-based
approach
that
that
Reichel
and
Hallberg
have
so.
We
have
CV
mix
in
there
and
we've
used
CV
mix
with
kpp
I
know
that's
what
ncar
is
using.
So
there's
no
more
bulk
mix
layer.
A
B
Okay,
thanks,
okay,
I
think
we'll
move
on
to
Aleister.
If
there
are
more.
Oh,
there
was
another
question:
joaning
go
ahead
and
unmute
yourself,
yeah.
F
Okay,
maybe
I
missed
it,
but
what
what
was
the
major
difference
between
the
the
two
vertical
coordinates
that
they
showed.
A
The
one
where
you
got
a
lot
of
heat
uptake
was
a
was
a
very
much
like
a
Geo
potential.
It
was
what
we
call
Zed
star
or
Z
star
in
in
American
parlance,
and
the
other
one
was
like
like
what
we
call
the
high
com
vertical
coordinate
where
the
interior
is
very
isopic,
no
like
in
the
upper
ocean
in
the
mix
in
the
boundary
there
is
more
Zed
star-like,
so
we
reduce
the
amount
of
need
to
re.
A
Well,
we
we
still
remap
the
same
amount,
but
the
the
amount
of
remapping
itself
is
much
less
in
the
interior
when
you're
kind
of
following
the
flow
itself
with
an
isopyolite
coordinate.
So
that's
the
only
difference.
Everything
else
is
the
same.
The
four
scene
protocol
is
the
same.
The
physical
parameterizations
are
the
same.
F
Okay,
so
let's
say
that
we
need
the
enough
amount
of
remapping
to
in
order
not
to.
A
Well
again,
again,
just
to
be
sure
we
understand
both
of
those
models
do
the
same
vertical
LaGrange,
your
remapping
algorithm.
It's
just
the
choice
of
vertical
coordinate
within
that
algorithm
that
differs
okay,
one
of
them
is,
is
100
Z
star
and
the
other
one
is
a
hybrid
Zed
star,
isopicnol,
okay,
okay,
so
yeah
cool.
B
Okay
out
pair
go
ahead
and
unmute
yourself.
C
So
I'm
wondering
if
the
re-gridding
could
be
made
adaptive
similar
to
Adaptive
mesh
refinement
where
you
can
apply
regretting
based
on
some
automated
quality
measure,
and
then
that
would
probably
also
allow
you
to
apply
re-gridding
locally,
both
temporally
and
spatially.
Would
that
be
possible.
A
D
So,
but
if
I
can
just
follow
up
so
there
was
a
student
of
anti-hogs
Angus
Gibson,
who
actually
did
an
Adaptive
isopenol
approach,
which
was
trying
to
optimize
where
to
put
surfaces
to
try
and
track
neutral
services,
and
then
Hans
bashard
has
similar
sort
of
ideas
going
on
for
Coastal
modeling,
so
I
mean
it's.
It
is
very
much
the
sort
of
thing
you
can
do.
There's
nothing!
There's
no
restrictions
on
what
the
coordinate
system
you
can
use
so.
A
I
think
one
of
the
one
of
what
Reiner
black
likes
to
use.
He,
he
I
think
he
coined
the
term
coordinate
free
I
mean
you
really
it's
just.
You
just
need
an
algorithm
to
tell
you
what
your
coordinate
surface
is
and
then
you
remap
to
it
and
that
that
algorithm
doesn't
need
to
be
written
down
in
a
in
a
form
of
a
pde.
It's
just
some
algorithm.