►
From YouTube: Robert Hallberg 2020 04 27
Description
No description was provided for this meeting.
If this is YOUR meeting, an easy way to fix this is to add a description to your video, wherever mtngs.io found it (probably YouTube).
A
Great
thank
you
Frank,
so
today,
I'm
going
to
spend
a
little
bit
of
time,
stepping
everybody
through
the
the
time
stepping
algorithms
in
the
mom6
ocean
model-
and
this
may
seem,
like
kind
of
a
strange
place
to
start
by
describing
the
model.
But
this
is
really
building
on
some
of
the
talks
that
hopefully,
you
saw
in
two
weeks
ago,
when
Steve
talked
about
the
arbitrary
lagrangian
eulerian.
A
Lagrangian
Dynamics
that
are
used
in
in
mom
6.
and
the
the
time
stepping
is
really
a
way
to
kind
of
think
about
how
the
the
model
as
a
whole
is
broken
up
into
into
bite-sized
pieces
and
hence
places
where
people
could
think
about
contributing,
perhaps
next
slide.
So
this
will
be
slide.
Two
Frank,
please
okay,
Okay
so
good.
So
the
Mountain
Six
ocean
model,
as
I
think
many
of
you
know
is-
is
a
community
ocean
model
very
much
rooted
in
the
world
of
climate
modeling?
So
it
is
deliberately
conservative.
A
It
is
designed
to
run
stably
for
for
many
centuries.
It
is
designed
to
work
in
all
parts
of
parameter
space
that
it
ever
finds
itself
in.
Hopefully,
but
one
of
the
things
I
think
is
is
really
important.
That
we
really
want
to
emphasize.
Is
the
community
open
development
of
mom
six?
This
is
moved
from
something
where,
in
the
past
you
would
have
say
a
gatekeeper
at
gfdl
who
might
have
to
sign
off
on
everything
so
something
where
we
have
have
Collective
contributions,
and
this
is
going
forward.
A
I
think
one
of
the
sources
of
the
great
strength
of
mom
six
and
the
community
behind
it,
in
that
we
can
draw
upon
expertise
in
a
wide
range
of
areas,
thinking
about
specific
processes,
thinking
about
how
to
tie
and
buy
a
geochemical
processes,
and
even
hopefully,
after
today's
talk.
Some
of
you
will
be
motivated
to
think
about
contributing
to
the
time
stepping
algorithm
itself
now
I
think
that
Steve
talked
last
week
about
some
of
the
nice
properties
of
the
arbitrary
branchian
eulerian
method,
as
used
in
in
mom
6
and
of
particular
relevance.
A
This
gives
you
access
to
General
vertical
coordinates,
basically,
anything
that
you
can
describe
as
a
coordinate
that
makes
any
sense
is
probably
achievable
within
the
code
and
for
many
of
the
purposes
for
climate.
One
of
the
critical
considerations
is
that
there
is
no
vertical
CFL
limit
on
the
vertical
infection,
so
people
who
used
older
models
like
pop
or
mob5
know
that
if
the
flow
got
too
energetic
the
way
the
model
would
would
die
is
that
the
vertical
advection
would
get
to
be
too
large,
and
then
it
would
bring
down
the
model.
A
Well,
the
fact
that
the
coordinates
themselves
are
moving
and
we're
remapping
through
the
entire
water
column
is
that's
not
not
a
way
that
the
model
can
itself
die,
but
a
lot
of
that
hinges
upon
the
underlying
design
of
the
algorithms
themselves.
So
if
we
go
to
the
next
slide,
please
Frank.
A
To
give
you
an
idea
of
how
these
pieces
fit
together
and
how,
in
fact,
they
emerge
from
the
underlying
nature
of
the
equations
that
we're
looking
at
and
then
I'll
spend
some
time
talking
about
a
couple
of
pieces
that
I
think
are
are
worth
thinking
about
more
broadly,
first,
the
Tracer
advection
strategies
that
are
used
in
in
mom
6
how
it
is
that
we
can
believe
we
can
take
long,
Tracer
time
steps
even
as
we
push
to
very
high
resolutions
and
then
secondly,
talking
some
about
the
barotropic
Bear
Clinic
split,
explicit
time,
stepping
used
within
mom
6,
which
is,
is
really
at
the
heart
of
understanding.
A
The
dynamic
core
of
the
model,
including
some
of
the
challenges
that
arise
from
the
use
of
a
non-linear
continuity,
solver
and
the
evolving
layer
thicknesses,
and
how
we
solve
these
in
mom
6..
Now,
people
who
have
used
models
like
pop
or
or
Mom
5
think
of
the
continuity
solver
as
just
a
simple
way
of
linearly
determining
a
vertical
velocity,
whereas
in
mom
six,
you
could
really
think
of
the
continuity
solver
as
really
being
at
the
heart
of
the
whole
algorithm,
even
more
so
than
the
momentum
equations.
A
Because
that's
the
piece
where
we're
enforcing
consistency
across
modes
and
where
a
lot
of
the
real
magic
in
this
happens
and
then,
finally,
for
those
Intrepid
enough
to
hang
on
through
the
entire
technical
details.
I
have
a
few
ideas
at
the
end
of
some
areas
where
I
think
the
mom's
takes
time.
Stepping
and
the
solver
might
actually
be
improved.
A
And
the
idea
here
is
we're
not
just
trying
to
give
people
an
idea
of
what's
done,
but
we're
also
trying
to
encourage
people
to
to
get
involved
and
suggesting
some
ways
that
that
people
might
think
about
it
in
some
applications
where
these
might
make
a
difference.
Next
slide,
please
Frank
right.
A
So
the
arbitrary
lagrangian
eulerian
method
involves
thinking
about
the
the
equations
that
are
satisfied
by
the
ocean
as
being
something
that
can
be
treated
in
two
separate
phases.
The
first
is
a
lagrangian
dynamic
update.
These
equations
are
highlighted
in
blue
in
the
momentum,
continuity
and
Tracer
equations,
and
they
are
essentially
the
shallow
water
equations.
A
These
are
all
things
that
do
not
require
the
infection
of
tracers
at
a
slightly
longer
time
scale.
There's
a
vertical
remapping
part
of
this
algorithm,
and
these
are
the
terms
highlighted
in
red,
and
these
are
completely
arbitrary.
How
you
do
this
and
dependent
on
your
vertical
coordinate
and
to
illustrate
how
these
two
pieces
fit
together,
there's
a
kind
of
a
schematic
on
the
side.
This
is
the
side
view
of
an
overflowing
gravity
current.
A
We
allow
the
coordinates
to
be
deformed,
and
this
is
how
we
capture
gravity
waves
and
the
restoring
pressure
forces
associated
with
them.
Immediately.
After
this,
we
will
remap
this
back
to
a
z,
coordinate
to
make
these
lines
completely
flat,
but
we'll
do
that
without
changing
the
state
of
the
model.
A
So
when
we
take
the
equations
themselves
into
consideration,
we
have
four
separate
timestamping
cycles
that
we
use
in
mom.
Six
that
routinely
use
different
time
steps.
The
first
one
in
black
shown
with
the
little
arrows
on
the
side
and
the
equations
are,
is
the
barotropic
time
steps.
Now
these
equations
have
been
vertically
averaged
for
the
momentum
equations
and
they
also
involve
the
the
a
vertically
integrated
continuity
equation
for
the
free
surface
height,
including
some.
A
We
have
the
lagrangian
Dynamics,
the
blue
equations,
which
are
essentially
the
three-dimensional
stacked
shallow
water
equations
they're
more
complicated.
They
include
not
all
the
non-linear
terms
that
you
have
in
the
the
momentum
equations.
They
include
pressure
forces
and
stresses,
on
the
right
hand,
side
momentum,
adduction,
in
this
case
it's
written
in
Vector
invariant
form,
and
then
you
have
a
layer-wise
continuity
equation,
where
the
layer
thicknesses
are
evolving
due
to
the
convergence
of
of
flow
within
the
individual
layers.
A
On
a
longer
time
scale.
We
can
average
over
multiple
of
these
lagrangian
Dynamics
time
steps
to
get
out
to
our
Tracer,
advection,
thermodynamics
and
and
vertical
mixing
time
steps
or
you
could
think
of
this
as
column
physics,
if
you're
coming
from
an
atmospheric
background-
and
this
would
include
things
like
evaporation
and
precipitation.
But
it's
also
including
vertical
mixing
turbulence
and
advection
of
temperature
and
salinity
in
the
horizontal.
A
Typically,
the
way
we'll
treat
this
is
will
either
treat
the
thermodynamics
at
the
start
of
of
the
step
or
at
the
end
of
the
step,
and
then
we'll
use
the
accumulated
Mass
fluxes
from
the
lagrangian
Dynamics
from
the
blue
cycle
to
step
the
Tracer
equations
forward
and
then
finally,
the
outermost
cycle
is
the
remapping
and
the
coordinate,
restoration
and
I've
depicted
this
as
as
arrows
just
going
horizontally,
as
opposed
to
looping
vertically,
because
these
are
really
something
you
should
think
of
not
as
a
continuous
process
in
time,
but
a
periodic
restoration
from
one
state
to
exactly
the
same
state
just
discretized
differently.
A
So
your
your
new
thicknesses
are
basically
where
you
specify
what
you
want
your
coordinate
to
be,
and
that
can
be
essentially
arbitrary.
Although
you
may
run
into
some
problems
in
the
blue
step.
If
the
coordinate
surfaces
are
are
too
kind
of
jumbled
and
entangled
from
one
column
to
the
The
Neighbors,
but
then
once
you've
specified
that
how
you
do
the
vertical
remapping
is
is
a
separate
piece
where
you
reconstruct
the
old
properties
and
then
integrate
them
over
the
new
finite
volume
cells.
A
There's
actually
a
fifth
time
step
that
can
fit
into
this,
which
is
the
time
step
with
which
we
do
forcing
from
the
atmosphere
and
that
may
occur
within
some
of
these
steps.
You
might
do
remapping
every
several
external,
forcing
time
steps
or
you
can
do
several
of
this
remapping
time
steps
per
forcing
time
step.
That's
entirely
up
to
you.
A
Okay,
next
slide,
please
think,
okay,
so
it's
worth
spending
a
moment
or
two
to
think
about
how
we're
doing
the
advection
of
scalars
of
tracers
in
mom
six,
and
for
this
we
are
routinely
using
the
piecewise
parabolic
method
or
a
optionally
a
simpler
variant,
and
the
idea
here
is
really
straightforward.
A
If
we
go
ahead
to
the
next
slide,
which
should
be
slide,
number
nine
I'm
skipping
over
seven
and
eight,
so
the
piecewise
parabolic
method,
you
can
uniquely
specify
a
parabolic
description
within
a
cell,
and
this
is
a
well-known
technique
from
several
papers
going
back
to
the
80s
and
90s,
just
based
on
the
cell
average
properties
and
the
and
the
edge
values
on
either
side
of
the
cell.
A
This
is
by
construction.
Conservative
approach,
you're,
guaranteed
that
the
cell
average
is
the
cell
average
and
since
you're
doing
the
integral
under
the
cell,
you
will
always
have
all
properties
accounted
for,
and
then
what
you'll
do
is
in
order
to
ensure
positive
definiteness,
you
can
adjust
the
left
or
the
right
Edge
values
independently
for
of
the
adjacent
cells
to
flatten
the
curves,
so
that
there
are
no
extrema
within
the
cell.
A
This
gives
monotonicity
in
one
dimension,
but
there's
a
challenge.
If
you
go
to
the
next
slide,
Frank,
please
Ted,
there's
a
challenge
in
ensuring
that
you
have
positive
monotonicity
in
in
multiple
dimensions,
and
the
the
idea
here
is
is
based
on
an
idea
that
originally
goes
back
to
paper
by
Russell
Lerner.
A
But
it's
it's
written
in
a
way
that
really
obscures
this
kind
of
critical
idea.
I,
like
subsequent
paper
by
by
Richard
Easter
from
month
weather
review
in
1993,
and
then
it's
also
described
in
Duran's
textbook,
where
for
each
of
the
different
directions,
we're
going
to
treat
the
fluxes
first
in
One,
Direction
and
then
the
other,
but
we're
going
to
allow
the
cell
volume
to
be
deformed
by
the
flows
that
follow
that
are
carrying
the
tracers
along
and
so
after.
A
Each
pass
in
One
Direction,
we'll
reconstruct
the
cell
average
properties
based
on
the
integrated
amount
to
the
Tracer.
That's
that's!
In
a
cell.
This
is
the
the
line
kind
of
on
the
right
column,
labeled
monotonic,
divided
by
the
the
volume
of
the
cell,
and
that
will
give
us
properties
that
are
monotonic
because
they're.
A
Basically,
the
averages
over
those
cells
that
were
we
were
doing
in
the
in
the
piecewise
reconstructions
and
then
we'll
do
the
convergence
or
Divergence
of
the
flow
in
the
other
direction,
based
on
fluxes,
using
these
updated
Tracer
concentrations,
but
still
cast
in
flux
form,
and
this
gives
us
a
scheme
that
is
guaranteed
to
be
monotonic
while
still
breaking
things
up
into
multiple
different
pieces.
A
There
are
some
other
ideas
that
that
have
been
used,
but
this
is
one
we
found
to
be
really
quite
effective.
There
are
some
mild
directional
splitting
errors.
If
you
really
care
about
it,
you
can
alternate
the
order
of
the
X
and
the
Y
directions
to
increase
the
order
of
accuracy
to
second
order
accuracy,
but
for
the
ocean,
where
flow
speeds
tend
to
be
relatively
small
compared
to
the
wave
speeds
that
set
the
the
time
steps.
A
We
really
haven't
found
that
this
directional
splitting
makes
that
big
of
of
a
difference-
and
this
is
something
that
is
absolutely
guaranteed
to
be
monotonic.
In
other
words,
you
will
not
create
any
new
extrema
in
any
Tracer
properties,
based
on
a
set
of
strictly
one-dimensional
monotonic
evaluations
of
the
fluxes
and
and
Mom
six
we're
using
a
piecewise
parabolic
method,
and
this
works
pretty
well
now.
A
There
are
a
couple
of
things
to
note
if
we
go
to
the
next
slide,
please
Frank,
okay,
so
the
first
thing
is
that
this
flux
form
pseudocompressibility
advection,
is
based
on
the
accumulated
mass
or
volume
fluxes,
not
on
the
actual
velocity.
So
all
you
have
to
keep
track
of
is
how
much
water
is
moving
in
and
out
of
the
faces
of
the
cells
and
how
much
is
staying
in
the
cells.
A
But
this
actually
has
some
really
powerful
consequences.
A
The
first
of
these
is,
if
you
note
that
the
layered
Dynamics
time
step
the
Blue
Cycle
from
before
those
time,
steps
are
limited
by
the
Doppler
shifted,
internal
gravity,
wave
speeds
or
the
inertial
oscillations,
in
other
words,
some
non-linear
inertial
gravity
waves,
but
in
the
ocean
most
of
the
actual
flow
speeds
tend
to
be
much
smaller
than
the
peak
internal
wave
speeds
and
I've
Illustrated
this
by
deliberately
taking
out
a
surface
speed,
but
a
speed
from
about
2500
meters
depth,
and
this
is
from
the
om4
model
that
Alistair
will
be
talking
about
later,
and
you
can
see
the
typical
speeds
in
the
interior
ocean
here
are
a
couple
of
centimeters
a
second,
whereas
the
fastest
X
internal
gravity,
wave
speeds,
maybe
you've
ordered
Doppler
shifted
internal
wave
speeds.
A
Maybe
you've
ordered
two
meters
per
second,
so
you'll
have
a
fairly
large
ratio
of
maybe
something
like
10
or
20
or
100
to
1
in
these
typical
speeds.
So
we're
really
looking
at
advective
fluxes
that
are
are
small
most
of
the
ocean
compared
to
the
speeds
associated
with
with
the
waves,
and
so
it's
very
convenient
to
accumulate
these
fluxes
or
multiple
Dynamic
time.
A
Steps
and
they're
still
in
the
limit
of
fairly
small
CFL
numbers
based
on
the
effective
speeds
now,
in
order
to
accommodate
those
few
places
where
the
speeds
are
fast,
we
add
in
extra
passes
of
Tracer
reduction
to
Mom
6
as
needed
in
those
small
fractions
of
the
cells
where
the
mass
fluxes
exceed
what
can
be
accommodated
within
a
single
pass.
This
may
be
associated
with
wetting
and
drying,
or
it
may
be
associated
with,
say
western
boundary
currents
where
the
flows
are
relatively
small.
Now
we
use
several
iterations.
A
You
can
estimate
how
many
you'll
need
beforehand,
but
we're
communicating
internally
to
accommodate
those
iterations.
But
what
this
really
gives
us
is
the
ability
to
take
much
longer
Tracer
time
steps
than
we
use
for
the
the
Dynamics.
And
so,
as
we
push
the
higher
resolutions,
we
think
we
can
hold
the
Tracer
time
steps
Associated
to
those
associated
with
resolving
the
diurnal
cycle,
even
as
we
push
to
finer
horizontal
resolutions
and
that's
one
of
the
big
powerful
things
as
we
go
forward
in
Tracer,
Rich
biogeochemical
models
next
slide,
please
Frank.
Okay,
we're
on
12th.
A
We
are
on
12.,
so
the
other
thing
that
I
think
is
worth
spending
a
little
bit
of
time.
Talking
about
and
I'll
go
through
this
a
little
bit
quickly,
because
there
are
a
lot
of
equations
here
and
the
equations
are
mostly
there
for
your
reference
I'm
going
to
not
dive
into
the
individual
equations
themselves
in
any
link.
A
That
is
how
we
do
the
split
time
stepping
of
the
barotropic
mode-
and
this
is
something
in
in
mom
six
and
in
notion
models
generally,
and
so
these
are
taking
into
account
the
fact
that
the
external
gravity
waves
are
typically
about
100
times
faster
than
the
Doppler
specific
internal
weights,
in
other
words,
the
time
scales
of
the
black
cycle.
This
rapid
cycle
are
associated
with
200
meter
per
second
external
waves,
as
opposed
to
two
meter
per
second
Peak
Flows,
In,
say
western
boundary
currents
and
the
way
that
we
we
do
this
is
we.
A
We
simplify
the
equations
to
a
two-dimensional
set
of
equations.
Now,
importantly,
in
doing
this
average,
this
is
the
average
accelerations
as
they
relate
to
the
movement
of
the
free
surface,
and
so,
if
you
think
well,
first
there's
the
pressure
due
to
the
free
surface
itself.
But
if
you
have
a
stratified
column
of
water
as
that
water
is
heaving
due
to
the
external
waves.
A
The
internal
stratification
is
also
contributing
to
through
the
pressure
gradient
forces
and
that's
something
that
is
important,
that
we
take
into
account
in
this
effective
gravity
that
I
have
listed
next
to
the
simplified
equations.
So
that's
not
the
just
the
gravity
of
the
presurface.
It's
also
due
to
the
interior
stratification
and
then
the
integrated
continuity
equation,
the
the
next
one
down
below
the
equation.
Next
to
the
simplified
equations.
A
This
is
an
integrated,
nonlinear
continuity
equation
and
it
allows
for
potentially
wetting
and
drying,
and
the
idea
here
is
that
you'll
use
a
two-dimensional,
solver
you'll
iterate
It
Forward
over
a
paraconic
Time
step
with
many
barotropic
time,
steps
of
order,
a
hundred
take
the
average
accelerations
from
that
barotropic
time,
step
subtract
off
the
vertically
averaged
accelerations
that
went
into
it
and
then
that
allows
you
to
advance
the
momentum.
Equations
and
the
layered
continuity
equations
over
a
much
longer
time
step
than
you
would
have
to
take.
A
If
you
tried
to
resolve
the
external
gravity
waves
now,
one
of
the
challenges
can
be
depicted
on
the
next
slide:
13.,
okay!
Well,
one
of
the
earlier
attempts
to
do
this
very
Clinic,
barotropic
split
time.
Stepping
is
is
described
in
a
nice
paper
by
Reiner
black
and
Linda
Smith
in
1990.,
in
which
they
only
took
into
account
the
external
mode.
When
you
do
that,
the
magnitude
of
the
eigenvalue
is
associated
with
these
two
modes.
A
So
there
are
actually
Four
eigen
eigenvalues
here,
but
as
I
adjust
the
length
of
the
time
step
compared
to
the
very
Clinic
mode
wave,
speed
and
move
to
the
right
on
this
curve,
you
can
see
that
I'm
picking
up
eigenvalues
that
have
this
kind
of
funny
pattern,
where
they're
growing
slightly
and
then
coming
back
down
the
other
one
is
flat,
and
so
wherever
this
eigenvalue
is,
is
bigger
than
one.
This
is
a
case
of
exponential
growth
and
in
instability
in
this
time,
stepping
scheme,
and
so
in
their
case
because
they
hadn't
included
the
interior
stratification.
A
They
were
leaving
out
part
of
the
external
wave
speed
in
a
way
that
was
leading
to
an
instability
that
was
recognized
by
Hickman
and
Bennett
about
1990.
five.
A
You
can
come
up
with
schemes
that
actually
capture
all
of
the
past
motions
in
the
barotropic
solver
and
if
you'd
like
you,
can
introduce
some
some
damping
as
well
of
the
very
Clinic
mode
which
can
be
helpful
because
here
you're
looking
at
high
frequency
waves,
that
are
probably
not
the
signal
that
you're
most
interested
in
and
may
may
give
you
some
trouble
with
the
with
the
the
time
stepping
scheme.
A
This
is
all
described
in
a
paper
that
I
wrote
now
20
some
years
ago
in
the
Journal
of
computational
physics,
and
this
time
stepping
scheme
for
the
two
layer
system
still
describes
what
we're
doing
in
in
in
mom
six.
Now
it
works.
Well,
it
is
demonstrably
stable,
but
there
are
other
more
accurate
schemes
out
there,
and
this
is
a
comparison
now
if
we
go
to
slide
15.
A
Yeah
I
might
have
I
might
have
jumped
over
one
of
these
slides,
okay,
so
the
scheme
on
the
left
shows
the
change
in
the
complex
phase,
speed
for
Bear
Clinic
time
steps
with
the
mom-6
time
stepping
scheme
the
scheme
on
the
right
describes
a
a
very
elegant
scheme
from
Ninja
McWilliams,
that's
used
in
ROMs.
In
both
cases.
What
you'd
like
to
see
is
that
the
the
red
lines
the
numerical
Solutions,
are
staying
close
to
the
unit
unit
circle
and
the
phase
change
the
distance
around
the
unit
circle.
A
They're
moving
from
the
the
axis
on
the
right
should
be
changing
by
the
right
amount.
Mom
six
is
using
a
a
quasi-second
order.
Second
order,
runga
scheme
for
inertial
oscillations
and
a
forward
backward
scheme
for
pressure
for
Gravity
waves.
A
The
ROM
scheme,
by
contrast,
is,
is
fully
third
order,
accurate
for
these
linear
and
visit
waves,
but
it's
using
Adam's
dashboard
techniques,
in
other
words,
it's
using
information
about
the
ocean
state
from
previous
time,
steps
that
are
not
available
to
us.
When
we're
doing
this
remapping
step,
we
basically
can't
carry
lots
of
information
from
previous
steps
and
interpolate
them
into
the
full
past.
The
vertical
remapping
in
any
kind
of
a
clean
way,
so
the
point
here
is
the
scheme
we've
got
is
one:
that's
that's
reasonably
efficient.
It
has
some
phase
errors.
A
As
you
can
see
here,
there
are
more
accurate
schemes
available,
but
it's
also
one
that's
carefully
designed
to
be
kind
of
more
on
the
Wanga
cut
a
family
of
time,
stepping
schemes
that
are
appropriate
for
an
ale
type
model.
If
anybody
is
really
ambitious,
I
suspect
that
there
may
be
better
approaches
out
there.
A
That
may
be
more
accurate
or
perhaps
more
efficient
approaches,
and
this
might
be
something
for
the
mathematically
inclined
to
to
really
dive
into
okay,
so
everything
I've
described
so
far
with
the
time
stepping
scheme
really
is
looking
at
the
linear,
say
two
layer
sort
of
equations,
but
it's
the
non-linearities
that
are
also
important
in
getting
something
that
will
work
well
in
practice,
and
the
challenge
here
is
that,
if
you're,
if
your
scheme
is
trying
to
solve
for
a
more
non-linear
problem,
in
other
words
where
your
layer
thicknesses
are
not
not
so
uniform
in
the
horizontal
or
where
they're
Vanishing,
either
in
time
or
in
space,
this
introduces
non-linearities,
and
there
are
actually
two
different
estimates.
A
A
If
you
go
on
to
slide
17,
these
are
described
in
in
somewhat
more
detail
how
these
evolved
and
that
the
problem
here
is
you
don't
really
know
what
the
vertically
integrated
thicknesses
are
at
the
velocity
points
until
you
know
what
the
velocities
are,
if
it,
if
you
have
a
linear
continuity
solver
as
you
would
in
pop
as
you
would
in
a
z,
coordinate
model
like
mom5
or
mitgcn.
Well,
this
is
pretty
trivial.
A
You
just
have
a
thickness
at
the
face
and
you
always
use
that
thickness
of
the
base,
but
Mom
6
is
using
a
piecewise
parabolic
method
to
reconstruct
the
layer,
thicknesses
and
in
exactly
the
same
way
that
we
were
doing
earlier
for
the
tracers.
We
now
have
an
effective
thickness
at
the
velocity
points.
A
That's
a
function
of
the
strength
of
the
flow
itself,
so,
if
you're,
drawing
from
a
a
thinner
cell,
that
effective
thickness
gets
thinner
if
you're
drawing
from
a
thicker
cell
you're
upwinding
the
estimate
of
thickness
that
thickness
gets
larger,
and
so
you
don't
know
what
these
thicknesses
are,
that
you
should
be
adding
up
for
the
barotropic
solver.
Until
you
know,
it's
velocity
is
going
to
be,
and
you
don't
know
that
velocity
until
you've,
Advanced
the
barotropic
solver
so
and
that's
I,
guess
the
point
that
I
was
making
in
slide
18.
A
You
now
have
two
separate
estimates
of
the
pre-surface
height
and
you
have
to
figure
out
how
to
deal
with
them
if
we
go
ahead
to
slide
19
the
one
of
the
earliest
proposals
for
how
to
reconcile
these
two
estimates
of
the
free
surface
height
is
the
one
proposed
by
black
and
Smith,
and
that
has
been
used
in
in
my
com
and
high
com
now
for
almost
30
years,
and
the
idea
is
pretty
simple:
you
simply
assume
that
any
differences
between
the
sum
of
the
layer,
thicknesses
and
the
sum
of
the
and
the
movement
of
the
free
surface
height
can
be
taken
into
account
by
simply
dilating
the
layers.
A
You
simply
add
up
the
one
add
up
the
other,
take
the
ratio
of
the
two
and
you
multiply
your
layer
thicknesses
by
the
movement
of
the
free
surface
height.
It
tends
to
be
relatively
small.
You
are,
after
all,
talking
about
having
movements
of
the
free
surface
height,
maybe
of
order
order
a
meter
or
so
and
you're
correcting
the
sum
of
the
layers,
and
so
they
can't
get
too
far
out
of
line
and
it's
really
quite
a
cheap
approach.
A
So
it's
very
appealing,
especially
if
you're
not
too
worried
about
about
conservation
and
if
you're,
using
the
the
model
for
say
a
30-day
forecast.
This
is
fine,
but
if
you're
doing
climate
runs
well,
a
climate
model
is
really
conservative
at
leading
order.
You
really
care
about
the
conservation
of
heat
and
salt
and
other
tracers.
The
total
mass
is
conserved
because
the
total
mass
was
conserved
in
the
barotropic
solver,
but
heat
salt
and
anything
else
is
not,
and
so
this
was
kind
of
recognized
as
as
a
a
bit
of
a
problem.
A
If
we
go
to
the
next
slide,
20
one
of
the
solutions
that
that
I
came
up
with
for
use
in
in
the
old
hymn
model,
the
predecessor
to
gold,
which
was
the
president
assessor
to
Mom
six
some
while
ago,
was
to
take
the
layers
as
the
right
answer,
but
then
add
in
a
a
fake
Mass
source
to
drive.
There
are
Tropic
equations
toward
the
layer
equations,
which
is
actually
having
the
effect
of
adding
a
flow
field
that
will
drive
the
layer
equations
towards
the
barotropic
equation.
A
But
the
problem
with
this
is
that
the
instantaneous
free
surface
Heights
from
these
two
models
can
be
very
gen
quite
different
and
you
can
generate
quite
energetic
external
gravity
waves,
and
this
is
Illustrated
on
the
next
slide
slide
20..
A
This
is
the
discrepancy
between
the
two
estimates
of
the
free
surface
height
at
some
time,
step
in
a
48
layer,
Global
model
with
a
one
hour
time
step
and
where
I'm
I'm
damping
the
two
towards
each
other
about
as
strongly
as
as
one
could
do,
but
you're,
seeing
that
the
sea
surface
height
discrepancies
in
in
some
strange
little
places
around
topography
and
especially
where
the
layer
structure
begins
to
break
down
around
Antarctica
can
be
some
tens
of
meters
and
the
typical
discrepancies
can
be
of
order,
a
meter
or
so
so.
This
is.
A
The
second
approach,
a
third
approach-
and
this
is
one
that
Paul
suggested
about
2003
and
Bob
Higdon
wrote
up
in
independently
in
2005-
is
to
take
your
two
estimates
and
then
say:
okay
well,
I
know
I
needed
to
move
water
from
one
cell
to
its
neighbors.
So
just
do
it
as
a
separate
upwind
corrected
step.
A
The
big
problem
with
this
is
that
when
you
do
this,
there's
a
tendency
to
move
the
densest
layers
up
slope.
So
this
leads
to
effectively
dense
water
tending
to
spontaneously
move
up
topography
cap
off
seamounts
and
to
lead
to
what
you
might
think
of
it
as
kind
of
a
Neptune
effect
circulation,
drawing
upon
Greg,
Holloway's,
old
old
ideas
of
some
rectified
effects
of
eddies,
but
essentially
you're,
adding
energy
to
the
the
problem,
and
it
leads
to
large
circulations
and
then.
A
Finally,
the
solution
that
we
we
came
up
with
alster
and
I,
going
back
about
10
years,
is
we're
going
to
take
the
continuity
equations
themselves
and
use
them
to
determine
what
added
philosophies?
What
added
barotropic
velocities
you
would
have
to
have
fed
into
the
continuity
equations
in
order
for
the
free
surface
Heights
to
exactly
agree
now.
A
The
continuity
equations
are,
of
course,
non-linear,
but
the
way
that
we're
getting
at
this
is
by
iteratively
determining
what
that
correction
would
be,
and
it
works
pretty
well
and
pretty
efficiently
and
part
of
the
key
to
the
success
here
is
that
the
iterations,
if
you
go
I'm
out
now
on
to
slide
number
24
I'm.
Sorry,
if
I'm
jumping
ahead
Frank,
the
the
continuity
scheme
itself
involves
the
construction
of
the
the
layer
thicknesses,
which
means,
even
though
the
thicknesses
themselves
can
have
some
structure
that
this
is.
A
The
transport
is
monotonically
increasing
and
smoothly
increasing
function
of
of
the
transport,
and
so
you
can
very
quickly
determine
the
right
transport
with
a
few
iterations
and
that's
Illustrated.
If
we
go
to
the
next
slide,
slide
number
25,
which
is
on
the
same
scale
as
the
one
I
showed
before.
But
these
are
the
sea
surface
height
discrepancies
after
three
iterations
to
try
to
determine
this
correct
velocity
and
the
the
inset
you
may
have
to
push
the
button
again
to
get
the
inset
if
there
isn't.
A
A
white
picture
of
a
PDF
shows
that
this
convert
that
this
convergence
is
is
really
quite
quite
fast
because
of
the
nature
of
these
piecewise
parabolic
distributions
of
the
thickness
so
with
just
a
couple
of
iterations
in
the
continuity
equations,
we're
basically
able
to
determine
exactly
the
flows
that
would
give
you
the
free
surface
height
that
the
barotropic
solver
said
you
should
and
the
transports
that
the
barotropic
solvers
said
you
should
have
if
you
go
to
the
next
slide
26.
This
is.
A
This
is
a
very
practical
approach,
in
the
sense
that
the,
although
the
thicknesses
on
the
top
left,
are
have
a
fair
amount
of
structure.
The
integrated
fluxes
are
are
pretty
nearly
linear,
and
so
what
we're
finding
is
that
this
converges
within
just
a
few
iterations.
The
other
thing
here
is
that
setting
up
the
sub
grit
scale
profiles
is
relatively
expensive
compared
to
evaluating
them
and
so
doing
this
iteration
in
the
continuity
solver
through
five
iterations
ads
in
in
one
particular
case,
the
one
I
showed
here.
A
A
Had
you
taken
short
time
steps,
so
the
fact
that
we
have
an
implicit
treatment
of
the
vertical
viscosity
and
we
have
a
no
slip
bottom
boundary
condition,
even
it's
over
a
relatively
small
thickness,
so
at
the
bottom,
given
by
something
maybe
like
the
square
root
of
the
viscosity
times
a
Time
step
or
or
something
like
an
Ekman
layer
depth.
A
We're
able
to
take
that
into
account.
So,
instead
of
applying
a
vertically
uniform
barotropic
velocity,
we
feed
the
continuity
equation
of
uniform,
barotropic
acceleration,
but
then
allow
it
as
we
see
on
the
next
slide.
Number
28
to
determine
the
the
transports
with
this
added
structure,
Associated
either
with
a
bottom
drag
or,
if
you
have
say
an
ice
shelf,
you
might
have
a
no-slip
top
boundary
condition,
and
this
is
something
that's
that's
working
quite
well
and
has
been.
B
A
For
about
about
12
years
now
in
in
gold
and
now
in
mom
six
now
there
are
a
couple
of
added
considerations.
If
you
go
to
slide
number
29
that
we
have
to
take
into
account.
This
is
I.
Don't
go
into
this
in
so
much
detail,
but
one
of
the
critical
points
is
that
we
don't
actually
feed
the
velocities
back
and
forth
between
the
into
the
barotropic
solver.
Instead,
it's
it's
the
transports.
A
The
integrated
transports
are
the
things
that
that
really
have
to
agree
in
order
to
get
the
gravity
waves,
the
external
gravity
waves
to
to
work
properly.
The
second
is,
and
again
this
is
a
little
bit
subtle,
but
everything
in
the
barotropic
solver
is
treated
as
anomalies
from
the
Bear
Clinic
state.
A
So
the
time
tendency
of
the
Velocity
is
this:
this
is
the
equation
about
two
thirds
of
the
way
down
are
are
given
by
the
anomalous
coriolis
accelerations
and
the
anomalous
pressure
force.
Accelerations
and
the
anomalous
drag
different
driven
flows
based
on
the
changing
barotropic
velocities,
the
changing
sea,
surface
Heights
and
then,
on
the
right
hand,
side.
We
have
basically
that
the
balanced
thickness
weighted
average
accelerations.
A
So
one
of
one
of
the
things
that
this
gives
us
is
is
that
if
you
take
this
split
paraclinic
barotropic
mode-
and
you
only
took
one
barotropic
concept
and
it
and
compared
that
with
the
model
where
you
didn't
do
the
splitting-
and
you
took
that
same
short
time
step,
you
get
the
same
answers
exactly
the
same
answers.
So
basically,
the
the
barotropic
solver
is
just
helping
us
to
get
past
the
the
time
step
limitations
for
these
rapidly
evolving
motions.
A
In
addition,
we
are
treating
both
the
the
bottom
and
the
surface
drag.
If
there's,
if
there's
say
an
ice
shelf
or
sea
Ice,
implicitly
in
the
barotropic
solver,
exactly
the
way
they
would
have
been
in
the
very
Clinic
solver,
and
we
are
also
by
by
virtue
of
the
fact
that
we're
approximating
the
transports
in
the
barotropic
equation,
as
kind
of
a
fit
to
the
sums
of
the
layers
we're
able
to
get
wetting
and
drying.
A
Even
in
this,
this
mode
split
form
in
a
conservative
and
physically
reasonable
way
in
the
interest
of
time,
I'm
not
going
to
go
into
the
details
on
on
slide
30,
which
which
basically
says
how
we're
doing
these
parametric
fits
of
the
transports
as
a
functions
of
the
varicopa
Velocity.
A
But
if
we
go
down
to
slide
31
the
takeaways
that
I'd
like
to
leave
you
with
regarding
the
mom6
split
time,
stepping
scheme
is,
although
it
may
seem
somewhat
complicated,
it
has
turned
out
to
be
a
very
efficient
approach
for
handling
the
fast
modes.
The
external
gravity
Waves
by
a
simplified
2D
equations,
giving
us
a
factor
of
something
like
50
speed
up
relative
to
what
you
would
do.
A
If
you
simply
took
the
three-dimensional
equations
and
stepped
them
with
a
short
time
step,
as
it's
in
fact
done
in
most
atmospheric
models,
there
use
a
noun
analog.
Excuse
me
is
an
analog
of
the
external
mode
in
in
the
atmosphere.
It's
known
as
lamb
wave,
don't
think
about
it
whole
month,
a
whole
lot,
but
it's
there
it's
detectable,
but
in
atmospheric
models.
There's
really
not
much
point
in
doing
it
this
mode
splitting,
because
that
external
mode
is
only
about
four
or
five
times
faster
than
the
fastest
flow
speeds
in
in
the
jet
stream.
A
You
don't
have
this
of
order
hundred
to
one
separation,
so
this
is
something
that
really
is
kind
of
unique
to
the
ocean,
but
common
across
a
variety
of
different
Ocean
Models.
A
The
barotropic
solver
is
the
piece
that
determines
the
evolution
of
the
pre-surface
height
and
the
time
average
depth
integrated
transports
in
mom
6
the
layers
accommodate
what
that
barotropic
solver
says
they
they
should
be,
but
we
have
actually
an
unsplit
time
stepping
scheme
available
in
mom
six,
and
you
can
use
it
to
verify
that
you
get
the
same
answers
if
you
take
a
short
time
step
and
one
barotropic
step
between
the
two:
it
works
with
wetting
and
drying,
and
a
lot
of
the
complexity
of
the
barotropic
solver
is
hidden
by
the
fact
that,
because
it's
linear,
we
can
automatically
tell
you,
for
instance,
exactly
what
the
longest
stable
time
step
would
be,
and
so
you
don't
have
to
think
too
much
about
it.
A
A
To
kind
of
sum,
this
up,
really
what
we've
done
is
developed
a
system
that
we
think
is
is
applicable
across
a
wide
range
of
different
examples,
and-
and
you
will
have
seen
many
of
these
examples
two
weeks
ago
and
in
some
of
the
applications
and
you'll
hear
more
from
from
Aleister
about
some
of
the
the
nice
properties
of
of
the
on4
coupled
Model,
A,
lot
of
which
are
derived
from
the
some
of
the
efficiencies
associated
with
the
time
stepping
scheme
and
with
with
the
fact
that
we
have
a
barotropic
solver.
A
That's
that's
working
quite
well
and
can
accommodate,
for
instance,
wetting
and
drying,
significant
movement
of
the
free
surface,
height
or
or
even
ice
sheets
and
Ice
shelves
and
going
forward.
We
think
really
significant
displacements
of
the
three
surface
Heights
due
to
things
like
large
tabular
icebergs.
A
So
the
one
last
thing
I
wanted
to
leave
you
with
on
slide
33.
Is
that,
although
I'm
presenting
all
of
this
as
though
it's
kind
of
a
fixed
package-
and
it's
all
working
well
I-
think
there
are
still
areas
that
could
be
grounds
for
improvement
and
they
can
be
coming
from
a
wide
range
of
different
areas.
So,
for
instance,
because
of
this
iterative
time,
stepping
of
the
tracers,
we
have
to
do
more
message
passing
than
we
think
we
might
have
to
do.
So.
A
Somebody
wanted
to
look
into
the
logic
behind
that
this
would
be
perhaps
one
area
that
we
could
work
on
to
make
the
model
scale
better
on
kind
of
a
computer
science
side
of
things.
On
the
other
hand,
the
way
that
we
do
the
Bear
Clinic
there
are
Tropic
time
stepping
scheme.
It's
a
predictor
corrector
scheme,
I'll
run
to
cut
a
two
type
scheme.
A
Well,
there's
some
speculation
in
conversations
that
Aleister
and
I
have
been
having
now
for
about
15
years
that
we
think
we
could
accommodate
all
of
this
with
a
single
step
scheme
which
might
lead
to
a
border
a
factor
of
two
speed
up,
because
you
wouldn't
have
to
run
through
the
barotropic
solver
twice
or
conversely,
for
forecasting
you
might
really
care
about
the
phase
of
those
high
frequency
waves,
in
which
case,
perhaps
a
higher
order.
A
Multi-Step
paraclinic,
barotropic
solver
might
be
suitable,
and
if
anybody
is
interested
in
that,
I
would
start
by
looking
at
Sasha
petkins
paper
where
he
goes
through
a
bunch
of
different
schemes,
most
of
which
are
not
appropriate
for
use
with
ale,
but
I.
Think
by
thinking
about
this.
There
might
be
ways
to
to
accommodate
this
and
make
things
work
better
for
those
particular
applications.
Whereas
we've
never
really
worried
about
this
for
climate,
because
we
don't
really
care
about
the
exact
phase
of
a
wave
and
what
it
looked
like
in
100
years.
A
For
for
climate-like
applications
thinking
even
more
broadly,
there
are
some
more
thinking
that
I
think
can
be
done
about
how
to
incorporate
the
fast
ice
Dynamics
into
the
barotropic
solvers
ways
to
avoid
coupled
instability.
This
is
something
I've
been
putting
some
work
on
in
the
last
couple
of
years
or
even
further.
There
are
some
places
where
we
haven't
carried
through
this
formal
linearization
that
works
so
well
for
the
American
barotropic
split
into
all
parts
of
the
model.
A
For
instance,
the
estimates
of
the
thicknesses
at
the
velocity
points
are
based
on
a
directionally,
biased
estimate,
but
it's
not
one.
That's
a
formal
linearization
of
the
continuity
solver.
So
there's
been
some
areas
there
where
there
might
be
room
for
for
improvement,
and
we
will
be
welcoming
of
anybody
hoping
to
contribute
by
thinking
about
any
of
these
things
so
I'll
leave
it
there,
Frank
I,
I,
hope
I
haven't
gone
too
much
over
the
time.
C
It's
okay,
I
think
it's
important!
This
is
information,
that's
not
easily
found
in
one
place,
so
it
was
great
having
it
summarized
in
in
your
talk,
I
think
if
there
are
some
very
quick
questions
of
clarification,
let's
take
those
now
otherwise
we'll
maybe
have
a
sort
of
a
broader
question
and
answer
period.
At
the
end,
any
quick
questions.
B
A
They
are
done
in
a
separate
step,
so
basically
we
do
them
implicitly,
wherever
possible.
Ex
sources
are
explicit.
Sinks
are
implicit.
This
is
important.
If
you
use
a
a
chemostat
equation,
you
can.
You
can
convince
yourself
that
this
gives
you
the
right
Solutions,
but
they're
done
with
a
a
backward
Euler
scheme
or
a
four-wheeler
scheme
which
formally
gives
you
first
order
accuracy,
but
it
gives
you
the
right
physical
Behavior.
A
If
somebody
were
ambitious
in
this
area,
knowing
something
about
the
rates
associated
with
the
sources
and
sinks,
I
suspect
that
you
could
come
up
with
a
non-linear
scheme
that
reverts
to
say
backward
Euler
for
a
sink
for
very
rapid
rates,
but
goes
to
say
a
trapezoidal
scheme
or
something
more
accurate
when
the
when
the
rates
are
very
well
captured
by
by
the
time
stepping.
But
it's
we're
using
kind
of
a
splitting
of
the
vertical
transport
vertical
mixing
with
sources
and
sinks.
Those
are
all
one
step
and
then
the
transport
is
separate
formally.
A
This
can
give
you
temporal
mode
splitting
splitting
errors
we've
played
around
with
using
string
splitting
say
doing
half
of
a
step
of
the
the
sources
of
full
step
of
detection
and
half
a
step,
and
it
hasn't
made
much
difference
in
in
the
cases
we've
looked
at.
But
those
are
all
things
that
could
be
relatively
easily
accommodated.