---
title: 'The Weather Equation - Numberphile'
source: 'https://youtube.com/watch?v=YmZiq00CO60'
video_id: 'YmZiq00CO60'
date: 2026-06-28
duration_sec: 2652
---

# The Weather Equation - Numberphile

> Source: [The Weather Equation - Numberphile](https://youtube.com/watch?v=YmZiq00CO60)

## Summary

The video explains the quasi-geostrophic omega equation, a fundamental diagnostic equation in meteorology that relates vertical velocity to advection of vorticity and temperature. It discusses the concepts of geostrophic balance, the Rossby number, and how forecasters use this equation qualitatively to diagnose weather development. The equation, though from early numerical models, remains a key conceptual tool for modern forecasters.

### Key Points

- **Introduction of QG omega equation** [0:30] — The quasi-geostrophic omega equation is presented: sigma d^2 omega + f^2 d^2 omega/dp^2 = f * vertical variation of advection of absolute geostrophic vorticity + advection of temperature gradient.
- **Diagnostic vs prognostic** [1:07] — This equation is diagnostic, not predictive; it diagnoses vertical velocity from current fields, not future states.
- **Vertical velocity and weather development** [2:46] — Vertical motion is crucial for development of low and high pressure systems; rising motion leads to low pressure, sinking to high pressure.
- **Omega as vertical velocity in pressure coordinates** [4:49] — Omega is vertical velocity in pressure coordinates (Pa/s), related to geometric vertical velocity W via hydrostatic equation: omega = -rho g W.
- **Absolute vorticity components** [10:31] — Absolute vorticity = relative vorticity (from shear and curvature) + planetary vorticity (Coriolis parameter f = 2*Omega*sin(latitude)).
- **Geostrophic balance** [16:12] — Geostrophic balance is a balance between pressure gradient force and Coriolis force, leading to winds parallel to isobars.
- **Rossby number** [27:19] — Rossby number Ro = V/(fL) quantifies geostrophic approximation; for synoptic scale Ro~0.1, for tropical cyclone eye Ro~1000.
- **Forecaster use of omega equation** [35:59] — Forecasters qualitatively assess vorticity advection and thermal advection to diagnose ascent/descent and development of pressure systems.

## Transcript

And this is one of the most famous
equations in meteorology. I'm just going
to go ahead and and write it down. So we
have sigma d^ 2 omega. So where the
omega comes from plus f 2 d2 omega by
dp^ 2. So this is the omega part and
that equals f * the vertical variation
of this term here which is the advection
of the absolute geostrophic vorticity by
the geostrophic wind and the advection
of geostrophic wind the temperature
gradient.
>> So this is this is the qua geostrophic
omega equation. So this is a fundamental
equation that allows us to through
systematic simplifications a lot of
simplification actually get the vertical
velocity from the adction of this
quantity called vorticity and the
adction of this quantity called
temperature. So thermal adction,
vorticity adection. And by looking at
horizontal maps of these adections, we
can diagnose vertical velocity and
vertical velocity is a proxy for
development. And development is your
developments of your lows, your
developments of your highs, high
pressure and low pressure. And that
relates to the weather forecast.
>> So what this tells you what's going to
happen in the future?
>> No. So that's quite an important
distinction about this equation. This
does this tells you nothing about the
future state of the atmosphere. So this
isn't a predictive equation. It's it's
it's called a prognostic equation. This
is a diagnostic equation. It just allows
us to diagnose the vertical velocity
from the invection of vorticity and the
invection of temperature.
>> And why is that an important thing to
know?
>> Um
>> if it doesn't tell you anything about
the future. Well, so this I mean we we
can go into this in a bit more detail uh
later on, but this um essentially is one
equation that falls out of a whole
system called the quai geostrophic
system. And I want to kind of go into a
bit about what we mean by quai
geostrophic omega. But essentially the
quai geostrophic system is a systematic
set of simplifications to the main
primitive equations which numerical
models these days use to forecast the
weather. um is a a systematic
simplification of those equations that
could then be used to make the first
kind of workable forecast models that
were used on the computers back in the
day the the early 1950s. And so this
equation is a diagnostic equation that
falls out of those and it is part of
that forecast process whereby you would
calculate some tendencies of various
things. You would then get the wind
fields which you could calculate wind
and thermal fields which you could
calculate the vorticity and the thermal
invection and from that you could
diagnose the vertical velocity and
getting to this vertical velocity is
actually quite difficult and so this
equation gives us a method to to
actually do that in a systematic and
relatively error-free way
>> in even more simple terms then this equ
is this equation relating to up and
downness of the air what's it like
what's what's it
>> exactly so when when we're talking about
vertical velocity We're talking about
the motion of the air in an up and down
sense. So on on large scales, so the
scales of weather systems, the main the
major motion of the atmosphere is in the
horizontal. So you've got winds north to
south, winds east to west. We don't tend
to think on large scales about the winds
that are going up and down, but they are
there. If you think about a convective
cloud, for example, they're moving up
into clouds, connects into clouds, and
you get rain falling out. Um, but on
these large scale, so we we're talking
about big scale weather systems here.
You know, systems the size of the UK,
systems that that that sort of take up a
big proportion of the Atlantic.
You don't tend to think about vertical
motions there, but they are there and
they're necessary in order to develop
the areas of low pressure and develop
the areas of high pressure. So if you
have rising motion, you get development
of low pressure at the surface. And if
you have sinking motion, then you have
development of high pressure at the
surface. One more basic weather question
before we, you know, get our teeth into
this. Then I'm pretty familiar with
measuring wind speeds horizontally. I've
seen those little devices like those
little cups that spin around and things
like that. Are there devices that
measure the up downwind, the vertical
wind? Like how do you how do you guys
take measurements of that?
>> No, it's very difficult. And that's one
of the reasons why we kind of look to
equations to to diagnose the vertical
velocity. It's very hard to measure.
it's on a much much much smaller scale
than the the horizontal motions. And in
fact, you can try and calculate the um
vertical motion from the horizontal
motions via something called the
continuity equation, which we'll have a
look at in a little while. Um but that
relies on you knowing to a very high
degree of accuracy your horizontal
winds. And it turns out that the the
divergence and the vertical motion
errors in that are the same size as the
actual vertical motion you're trying to
diagnose in the first place. So you can
have small areas in the wind that can
lead to massive areas in vertical
velocity.
>> Just quickly, what are the things that
measure the wind speed, Cole?
>> Anim
>> an animometers.
>> Animasure
wind speed horizontally.
>> Yes.
>> Are you saying equations the likes of
these are kind of what you have instead
of those for vertical.
>> That's how you get to the vertical
velocity.
>> Yeah.
>> So let's break it down. So um first of
all, it's the omega equation. So let's
talk about omega. So we got this Greek
letter omega here. So omega is just the
vertical velocity. So it has a
equivalent called W and these two are
equivalent but they use different
coordinate systems. So for example, if
we think about W, this might be a little
bit more familiar. We think about a
coordinate system that has winds in the
XY. So this is like your XY plane and
then you've got a vertical plane. If you
have some vertical motion or if you have
some horizontal motion, we could have a
a U wind in the X direction, a V wind in
the Y direction. So these are our
horizontal winds and then w would be um
the wind in the vertical and that would
be measured in meters per second. There
is a direct equivalent though uh in a
different coordinate system which we in
meteorology that we like to use because
it has some good advantages when coming
to equations like this and looking at
things like mass continuity. We still
have an xy plane but we use pressure as
our vertical coordinate. And so if we
have vertical motion in the pressure
coordinate and we can use pressure as a
coordinate because pressure always
decreases with height. So it's around
about a th00and millibars at the surface
1,000 hector pascals. At the top of the
troposphere which is the top of the
weather bearing layer is around about
300 hector pascals. So pressure always
falls with height. So if you're moving
in an upward direction you're moving
towards lower pressure. Ascent in the
the geometric coordinate system would be
W. ascent in the pressure coordinate
system would be negative omega because
you're going towards lower pressure as
you go up whereas in the geometric
system you're going towards kind of
higher elevations as you go up.
>> So does that mean we're using on our x
and our y axis here we're using
different units because that's
presumably still meters/s.
That's right. Yep. So this we still have
a U wind and a vwind in meters per
second
>> but we're using a different unit on
that.
>> Yep. The unit for this one would be
pascals per second. It's it's the amount
of pressure change that this parcel of
air is experiencing per second. So very
much like your your speed would be the
amount of displacement you're
experiencing per second, meters/s. This
would be the the pressure change per
second, pascals per second. Okay, cool.
You can you can link the two uh via
something called the hydrostatic
equation. So we have this hydrostatic
equation which relates the rate of
change of pressure in the z direction
with this term minus row which is the
density and g which is gravity. You can
say that um if you treat is the material
derivative which you can expand out to
this expression here. And um we're going
to assume something called a hydrostatic
balance. Um, and what hydrostatic
balance effectively means is that if we
have an atmosphere that's at rest and we
have a particle or a parcel of air in
the atmosphere, it's got two forces
acting upon it. We've already said that
pressure decreases with height. So,
we've got a pressure gradient in this
direction. So, this parcel has
experienced a pressure gradient force in
that direction. And then we've got a
gravitational force in that direction.
So, we've got pressure gradient force
and gravity. And when these two are in
balance, you know, if if there wasn't
gravity and there was a pressure
gradient force, then this parcel of air
would fly up into the atmosphere because
it'd be the the higher pre moving from
higher pressure to lower pressure and be
pushing it up to into the atmosphere.
But because it's balanced by gravity,
the the particle or the air parcel is at
rest and we call this hydrostatic
balance. So we assume the atmosphere is
at rest. There's no U, there's no V,
there's no pressure change with time. So
we can say that uh omega is
approximately equal to W DP by DZ. And
then from this hydrostatic balance
relationship which we kind of expanded
out here we can say that omega be equal
to minus row g which is this term here
times w. So what this effectively tells
us is that uh omega and w are related
but omega is just a scaled version
scaled by the density because because
the atmosphere is more dense at the
surface. So if you have a parcel of air
that's moving vertically at fairly low
elevations, the pressure is quite high.
Because the atmosphere is more dense
here, the pressure levels are kind of
closer together. And therefore the same
vertical speed in height coordinates at
lower levels would be crossing more. It
be the pressure will be changing more at
lower levels than it is at higher levels
for the same vertical speed when you're
thinking about the change in meters. So
this is what this equation tells us.
Omega is just a scaled version of the um
vertical velocity in geometric height
coordinates, but it's scaled by the
density to account for this effect.
>> Want another piece of paper?
>> Yeah, I think so.
>> All right.
>> The main takeaway about omega is it's
the vertical velocity in a pressure
coordinate system. Okay. So, there's a
few other symbols that we uh we want to
look at here. Yeah.
>> So again we'll just concentrate on the
left hand side at the moment. Right. So
this is sigma. Sigma scales the vertical
velocity response for a certain size of
forcing. This all taken together gives
the three-dimensional distribution of
omega or the distribution of the
curvature of omega. So it doesn't
necessarily give us omega itself yet but
we'll see how we can retrieve omega from
this side. But it tells us something
about the threedimensional distribution
of omega. Now these terms on the right
hand side let's break this down. So this
this is a za the G means geostrophic.
And I want to delve a little bit more
into what geostrophic means in a second.
But this Z to G plus F this tells us the
absolute vorticity. Vorticity is
effectively a measure of the spin in the
atmosphere. Any atmospheric process
where you've got things spinning around
any fluid spinning you have vorticity.
And these two different types of spin
that we've got make that make up the
absolute vorticity are the relative
vorticity. So relativeity is just the
spin. If you put a paddle wheel in the
flow, say you've got some flow coming
down here and you've got a lesser flow
here and you put a little paddle wheel
in the flow. Well, there's more pressure
on in the flow on this part of the
paddle wheel than there is on this part.
So this paddle wheel is going to start
to spin. So the fluid at this point
would spin in that direction. And that's
the vorticity. Now, this would be a
sheer vorticity because the we've got a
shear. Um, fluid passes are moving
faster here than they are here. You
could have a curvature vorticity where
simply the flow is curved and so if you
put a stick in a flow here, it would
curve. And so again, you've got
vorticity through through curvature. So
vorticity or spin can come out through
through different mechanisms. Uh, Z to G
here just tells us about the vorticity
of the flow. Now this f this is the
planetary vorticity and this is just
vorticity that you have even you have it
Brady just by virtue of being on a
spinning earth. So anything that's on
the earth apart from on the equator has
some kind of spin about its local
vertical. Yeah. If you think about the
globe it's spinning on its axis. So if
you were at the north pole you would be
experiencing that full planetary
vorticity the full spin of the earth.
You wouldn't be able you wouldn't know
it was there because you're kind of
spinning along with it. everything that
you can see is spinning along with it.
So you wouldn't know it's there, but it
is. It's spinning. And you can pick any
point on the Earth's axis and you can
decompose its spin um into a component
about its local vertical. Um and the
equation for that F is equal to 2 *
another omega but this is the big omega
s of the latitude. We calculate what our
planetary vorticity here um at the Met
headquarters is in exit. If we take this
equation here. So f which is our
corololis parameter is equal to 2 *
omega. Now omega from memory is
something like 7.292
* 10 - 5. This is in radians/s. So this
is just the the number of radians that
the earth is turning per second.
>> So that's a constant.
>> This is a constant. This is a constant.
The only thing that varies is the
latitude. And we take sign of the
latitude. Um I don't know what it is in
radians unfortunately. But because we're
taking a sign of it, I don't think it
matters too much. Uh sin 50°. And that
turns out to be, if you go through the
calculation, it's approximately uh 1.1 *
10 -4 per second. So that's the
planetary vorticity. So go back to the
equation. The relative vorticity of flow
plus the planetary vorticity, that's the
absolute vorticity. But this little
quantity here, this little dell symbol
means that we take the gradients of
that. Gradient means that the absolute
velocity is changing in space. We take
the gradient of that and we do a dot
product with the uh wind velocity and
that gives us the advection of the
gradient. So it's how the wind is
blowing along the vorticity gradient
kind of tells us you can imagine that's
the advection of the vorticity. If
you're going from an area of high
vorticity to low vorticity, you're
blowing the high volticity towards you
and therefore you got positive vorticity
in vection.
>> I'm beginning to get some appreciation
as to why weather is complicated. Well,
I that's a really good point. And and
when when people say when people say,
"Oh, weather forecasting is easy, isn't
it?" I just look out the window. Then my
temptation is to point them to stuff
like this and say, "Well, actually,
there's a lot more to it than that." Um,
so this is the vorticction part, and
we'll come on to kind of what that
physically means in a little while. Um
this is the temperature vection part. So
very much like we had the gradient of
temperature. So this is a bit more
simple. You can just think of this as
temperature. We we but we've got the
temperature gradient and again we're
vecting that with the geostrophic wind.
And these features in in front of all of
this well these are this is just the gas
constant. This is pressure. This ter
this little expression here means not
only are we taking the gradient of
absolute vorticity and advecting it with
a geostrophic wind. We're actually
taking the vertical variation of that
which is what this uh d by dp term tells
us and then it's scaled by the coralous
parameter f.
>> Even you know this is ridiculous.
>> Even you know this is ridiculous. All
right.
>> Yeah. So before we get to how we use
this equation as weather forecasters and
believe it or not given all the
complexities in a conceptual way, we
still do use this equation which is um
fascinating to me because if you think
about the quai gestroic equation set
which was derived to make the first
workable numerical models. Well, pretty
much all of the rest of that system has
not fall well, it has fallen out of use
as models have got better, super got
faster, but the one kind of enduring
thing that we use as forecasters is this
QG equation, which is why it's such a
famous equation of meteorology. Um, and
it's why it's so interesting to me. But
we're we're a couple of steps away, I
think, from kind of understanding what
this physically means. And I think we
need to go on to talk about geostrophe
and what geostrophic means and then we
can talk about what quai geostrophic
means. So should we have another piece
of paper?
>> Yes we shall.
>> We've talked about what omega means. Uh
I just want to spend a little bit of
time about talking what geostrophic
means and then by extension we can go to
quai geostrophic. We can go back to my
lovely rendition of the globe here. We
can imagine a weather system on this.
And you this might be sort of quite
familiar if you imagine this might be
like a typical weather map that you
would see with an area of low pressure.
And when you see the forecasts on the TV
and they show the the pressure maps,
they often show the winds. If you look
at how the winds are blowing, you find
that one, they blow counterclockwise
around an area over low pressure, and
two, they tend to blow parallel to the
isobars, which is quite surprising if
you think about it because if you think
you've got a region of low pressure,
that's a bit like a vacuum cleaner. And
if you think about what happens when you
turn on a vacuum cleaner, it sucks up
all the dust and grime and sucks up the
tube and it's creating an area of low
pressure and the wind is effectively
blowing in towards that vacuum. So why
doesn't it happen um on our planet? Uh
and the reason for this is because um as
well as the pressure gradient force
there's another force that we have to
take into consideration. So if we
simplify and we think of our air parcel
moving this is like the pressure
gradients of the same as on the map and
we'll just imagine we've got an area of
high pressure here and an area of low
pressure here. Now at the moment we
don't really know anything about the
direction of this but um it it will be
blowing in this direction. Um it's B's
ballot law actually which says if the
wind is in your face so if you were
standing here Brady and the wind's in
your face uh you would have low pressure
on your right
>> in both hemispheres.
>> Uh I have to think about that one. Um so
in the southern hemisphere the wind does
blow clockwise around an area of low
pressure. So if you were standing if you
face the wind it would be to your left.
Yes. It would be the opposite.
>> That's right.
>> Not that we're being hemisphere. Not
that we're not we're discriminating
against either hemisphere. We we're
hemisphere agnostic.
>> So we would have a pressure gradient
force in this direction and if that was
the only force acting then the air
parcel instead of going in this
direction would go towards the low
pressure. But it turns out that's
exactly balanced by this other force and
this is the corololis force. This is
well it's a little bit controversial as
to what you describe it. Some people
describe them as fictitious forces. Some
people will describe them as forces that
you have to include in the equations of
motion in order to make it appear as if
we're not on a rotating planet because
although we are on a rotating planet, we
can't really judge that. But the motion,
the fluid motion, the motion of the air
parcels knows we're on a rotating
planet. And so you see some odd
behaviors. And one of the odd behaviors
is if you have an air parcel that starts
to move from high pressure to low
pressure, it will feel this corololis
force acting and pulling it to the
right. And so as it moves, it will
continue to feel this force until
eventually and the pressure gradient
force is always acting in this
direction. And it will continue to be
pulled to the right until the corololis
force and the pressure gradient force
balance. And then we end up with this
what we call geostrophic flow. So
geostrophic just basically means there's
a balance between the corololis force
and the pressure gradient force. That's
not fictitious.
>> Well, the pressure gradient force isn't
fictitious, but the corololis force is
kind of fictitious because it's a it's a
correction that we have to add to
account for what we see. Corololis is a
really interesting character and one of
the roads around the Met Office is named
after him. It's of such fundamental
importance to weather prediction. We can
look at the equations of motion. Um, so
I'm going to write down the equations of
motion. Now you you've done the Navy
Stokes equations. These equations of
motion are sort of analogous to the to
those. And if we take the equation of
motion in the x direction. So this
relates the acceleration of the u wind.
So if you remember going back to our
coordinates, we had a u wind in the x
direction. So how the u wind is changing
with time is the acceleration in the u
direction and that's equal to this term
here which is the pressure gradient
force in the x direction. This is the
corololis force. And then we've got some
frictional terms. And this basically
boils down to F= ma Newton's second law.
So we've got our forces on one side got
our acceleration on the other side. If
we apply it to a parcel with mass one
unit mass then we end up with basically
A= F F= A. So this is Newton's second
law in a nutshell and I'll write just
quickly write out the one for the V
because it follows a very similar
pattern. So there's the pressure
gradient force in the Y direction. Corus
term is minus FU and then plus some kind
of frictional forces. When we looking at
geostrophic balance, um, we are assuming
first of all, we're assuming there's no
friction. So, we can't really apply
geostrophic balance at the surface. And
that is why when you see winds blowing
around an area of low pressure and you
look at the surface winds, they're not
quite parallel to the isobars. They're
actually just pointed in slightly
towards the low pressure. That's because
this frictional force is having a drag
effect that's pulling the winds in
towards the low pressure. So the
geostrophic approximation is is a good
first approximation for calculating the
winds but it's it is an approximation um
partly because of these frictional
effects but that's dragging along the
ground.
>> It's dra exactly that. Yeah it's
experiencing a stress against the
ground. It's losing it's losing
momentum. It's falling slightly down the
pressure gradient um rather than just
going around it and therefore eventually
the winds all kind of curling into the
the center of the low. And if there was
no process above the load to kind of get
rid of air or mass, then that low
pressure would just gradually fill up
and would become not a low pressure
anymore. So we can get rid of these
frictional forces because they're in the
geostrophic system, we consider them
negligible. And also we need to have a
non-acelerating atmosphere. So an
acceleration could be a speed up or a
slow down or it could be a curve because
anything that's moving in a curve is
being accelerated towards the center of
that curve. Anytime you introduce an
acceleration and change the strength of
the wind, you change this corololis
force term because corololis term is a
function of the wind speed itself. Is
the sun pumping a whole lot of heat into
the system not going to cause things to
speed up or accelerate?
>> So at the moment we're kind of we're
neglecting that we're the sun will have
uh an impact on smaller scales. So think
of that convective cloud we talked about
that cumulus cloud. Sun heats the ground
and introduces all kinds of motion. But
the the these the scales that we're
talking about, these very large scales,
the the direct heating input from the
sun um is
not so much of an effect. It obviously
has an effect. It creates the conditions
necessary to drive weather. Um but it's
not incorporated in this simplified
system. So yeah, anytime you introduce
an acceleration um you will disrupt this
force balance because the corololis
force is a function of the wind. So if
you change the wind, you change the
corololis force. These forces are no
longer in balance and therefore the flow
wouldn't be geostrophic. So basically we
just ignore the accelerations as well.
And this gives us our geostrophic
balance. So it essentially says this
pressure gradient force and this
coralous force balance. And I can just
explicitly show that by taking the
pressure gradient force over to the left
hand side of the x equation. And
similarly to the y equation and then we
can divide both sides by f. And this
gives us an expression for our
geostrophic wind on that diagram of the
earth. When I said you know you look at
the you look at the winds on the
forecast map to a first order
approximation or to a good order
approximation. The winds are
geostrophic. They blow parallel to the
isobars and you can calculate those
winds. So we're at a point now whereas
remember when you said um can you not
measure the the vertical velocity? Can
you not measure it? Well, we can measure
the horizontal winds but the problem and
calculate the vertical velocity from
that. But if you remember the problem
from that is that we we have slight
errors in our wind measurements that
measurement the wind measurements are
quite sparse. So we don't know what's
happening in between and the errors
introduced by those measurements dwarf
the vertical velocity itself. But we've
got a situation here where we could
actually if we just know the pressure
gradients and the corololis parameter
which we can calculate we can actually
calculate the winds. So we don't need to
measure them anymore. We can calculate
them exactly.
>> Where do you get the pressure gradients
from?
>> So measurement of pressure at the
surface you would just use an instrument
called a barometer. So that's the that's
the pressure kind of equivalent of an
anometer for the wind. So we've got a
barometer and an animometer. Measuring
pressure through depth. Um that's a
slightly different story. We can cover
that another time.
One of the reasons I wanted to just
describe geostrophic wind is because I
wanted to show you that you can a
connect the vertical motion to a
quantity which we call divergence but b
the geostrophic wind can't necessarily
help us with this right so I want to
take the um equation for mass continuity
du by dx plus dv by dy plus d omega by
dp equals not. So you've heard of the
principle of conservation of mass. This
is the meteorological equivalent of
this. And what it tells us is that if we
rearrange this, we say that the rate of
change of the u in the x direction. So
rate of change of the v wind in the y
direction is equal to minus the omega
dp.
Now we've got some quantity with omega
which is vertical velocity. And this
expression here is identical to
something which we call the horizontal
divergence
which is this product here. So
physically you just think about that as
air in a column. If the air is moving
apart out of that column you've got
divergence of the wind and that
generates through this equation some
kind of vertical motion.
>> Basically the all the energy of the wind
has to go somewhere has to do something.
>> It's not the energy it's the
conservation of mass. So if you're
removing the mass in the horizontal
direction, then you've got to accumulate
the mass from the vertical direction.
But the problem with the geostrophic
wind is we can calculate the divergence
of the geostrophic wind. If we plug this
into this equation here, so if we called
this du and dvg instead, then we could
take d by dx and d by dy of our
geostrophic winds. We can plug these
geostrophic equations in. So, du by dx d
by dx of this equation here -1 / row f
dp dy plus and then d by dy of min -1
over row f by dx plus one over row f.
That's that one there. And you can see
here we've got basically a dp a dx dy a
dp a dx dy. We've got a plus one over
row f and a minus one over row f. So the
two terms cancel out and that equals
zero. So this is quite an important
result that the divergence of the
geostrophic wind is zero. But we need
the divergence in order to say something
about the vertical motion. So this
brings us back to our quai geostrophic
omega equation. Yeah. All right. So we
need a new piece of paper.
>> All right. Let's do it.
>> Let's talk about the geostrophic system.
And one of the assumptions that is made
when you go from the full primitive
equation set to the quai geostrophic
equation set is you assume that the
atmosphere is in geostrophic and
hydrostatic balance. We can't assume
that it's completely in that way because
as I already showed you the geostrophic
wind is not divergent. So we can't get
vertical velocity from the just the pure
geostrophic winds. So that's where the
quai comes from. You do a systematic
simplification of the equations. You
have to retain the full wind in just
enough terms in order to be able to find
vertical velocity but remove it from as
much as possible in order to make it as
simple as possible. That's where quai
geostrophic comes from. But you might
ask how good an assumption is it to say
that for the types of weather systems
that we're considering so the areas of
low pressure, high pressure and the
fronts and the cloud and the rain. How
how relevant or how appropriate is it to
say that the atmosphere is in
geostrophic balance? And so I just want
to introduce this cool number called the
Rosby number. It's what we call a
non-dimensional number. So it doesn't
have dimensions and it's made up of a
series of variables, but it is a number
and you can use it to show things about
geostrophe and and and what have you and
the rest of it. So how do we get to the
Rosby number? So Rosby comes from KL
Rosby. He's a famous meteorologist from
the early 1900s. He's as far as I know
the only meteorologist to have ever
appeared on the cover of Time magazine.
Amongst many things, one of his claims
to fame, he derived this number called
the Rosby number which is essentially
the ratio of the acceleration of the
flow. So going back to our equations of
motion here, the acceleration of the
flow to the corololis acceleration. We
take the ratio of the acceleration of
the flow which in scale analysis terms
is so du by dt it's some kind of speed
scale over some kind of time scale. So
I'm using scales here because eventually
I want to think about well what's the
typical scale of our weather system and
plug and plug this in and get a rosby
number. So this this is our acceleration
and this is our corololis which is Fus
force F times acceleration equivalent F
times a speed scale and then we can just
simplify that to say V that's the same
as V over FTV the V's cancel you get 1
over FT it's not quite the quite the
rasby number in the form I want to show
you because what we want to do is take
the kind of speed equals distance over
time equation very simple so speed
equals distance so a length scale
divided by a time scale. I'm just going
to substitute t in for that. So that
gives t=
l / v. And we plug that t into there and
we end up with 1 / f * l over v. And we
basically just flip that upside down,
put it on the top, which gives us v over
l roby number. So this is a a
dimensionous number. If you plug in the
scale, so you got velocity in meters/s,
this is in per second, and this is in
meters. So you got meters/s over meters
per second. The dimensions cancel, but
it allows us to say something about
geostrophic flow.
>> Velocity of what? Of the wind.
>> In the meteorological systems, we're
talking about the velocity of the wind,
but it could be the velocity of wind in
a tornado with velocity of a wind in a
large area of low pressure. The length
scale for a tornado would be of order
100 meters. But the length scale for a
large scale weather system would be of
the order of a thousand kilometers. So,
um, this is the Rosby number. It allows
us to say, well, how geostrophic is our
flow? So if you just plugged in pure
geostrophic flow well the length scale
of pure geostrophic flow is infinite or
you could another way to look at it is
the acceleration is zero. So you end up
with zero over this roby number equals
zero. So for pure theoretically pure
gistrophic flow roby number is zero.
What we've shown here is that as the
Rosby number gets smaller and smaller,
zero is the smallest it can be, but as
it gets smaller and smaller, the the
more towards geostrophic the flow gets
because the corololis is more dominant
than the flow acceleration which matches
our corololis force here. So for a
synoptic scale system, you know, the
typical wind speed might be 10 m/s, you
know, 1 m per second too slow, 100
meters per second, too fast. So typical
might be 10 m/s. Typical length scale uh
might be of order 1 th00and km said. So
>> are you saying for all the all the
points along those thousand km the wind
is that speed.
>> Yeah. Well
>> because the wind speed is always like in
one position it's a certain speed.
>> That's right. Yeah. But we're thinking
about orders of magnitude here rather
than exact values at the moment. So when
thinking about the length scale I'm
think about over across entire low
pressure is about 1 th00and kilometers
across. So low pressures of 100 km
across that's quite small for an area of
low pressure. 10,000 km is far too big.
So th00and is about the order of
magnitude. So this is what we think
about orders of magnitude here. And we
already calculated that the f for our
kind of you know rough mid latitude
location is of order 1* 10us 4 which we
calculated earlier. If we plug them plug
these into the Rosby equation then we
can say that Rosby number for synoptic
scale flow is 10 / 10^ 6 * 10us 4 which
equ= 10 / 10^ 2 which is 10 over 100
which is 1 over 10 0.1
>> Rosby number 0.1. Okay.
>> Yeah. So the Rosby number for your
typical synoptic scale weather system so
your typical areas of low pressure and
high pressure is around 0.1. That's very
close to zero. So that tells us that to
a very good approximation the winds
around of low pressure and high pressure
can be assumed to be in geostrophic
balance which is this balance between
pressure gradient and coralis force.
>> So a roby number is like a little
waiting you add to tell you it's almost
like it's almost like an error bar or
how close you are to this ideal that you
wish you had that would make life easy.
>> You can yeah you can think of it like
that. Yeah. The smaller it is the closer
you are to geostrophic balance. And you
could bung in you could bung in for say
like um I don't know the eye of a
tropical cyclone
where you know the wind speed 100 meter
pers maybe a little bit too strong but
we'll we'll go with it. A typical length
scale of the eye of a tropical cyclone
maybe 10 km which would be 10^ the 4 m
and tropical cyclones tend to form at
lower latitudes. The corololis parameter
is maybe an order of magnitude smaller.
And so if we plung these into our Rosby
uh number equation, then we get V 100
divided by 10 4 * 10 - 5, which is 100
over 10 to the minus one over 10 equals
a,000.
>> Wow.
>> That that you can see is a massively
different roby number to a stomp car
system. So the type of balance we're
looking at when you're thinking about
the force balances in the tropical
cyclone eye is not geostrophic. It's
something we call cyclloic
um which is a different balance
altogether. But that's what's really
cool about this rby number. You can plug
in the scales of your system under
consideration and it quantifies how good
the geostrophic approximation is in your
system that you're looking at.
Right? So here's our qua geostrophic
omega equation. So we've talked about
omega that's vertical velocity. We've
talked about geostrophic that's some
kind of balance between pressure
gradient force and coralous force. Now
this is quai geostrophic because when
you go from the full primitive equation
set and you simplify it down using
systematic scale analysis and
simplification, you need to retain the
full wind in some of the terms in order
to produce vertical velocity as I showed
you through the continuity equation. So
you can't have everything geostrophic
because if you did you would remove the
information about the vertical velocity.
Um so that's what makes it quai
geostrophic. It's not quite geostrophic
but it's it's near enough.
>> Now is the Rosby number in here?
>> Rosby number is not in here. No no
unfortunately not.
>> Okay.
>> But it is a cool number. So when we
think about this, so the whole the whole
point of me explaining to this to you is
to get to the point where I can show you
how we as meteorologists use the
principles contained in this equation to
help us diagnose things about the
weather. Now to get that what we do is
we tend to partition these two sides
into something called the response and
the forcing. So as meteorologists we are
basically looking at weather maps of
vorticity and maps of temperature or or
thickness as we we like to look at it
and we look at those as the forcing
terms and it's this that's forcing a
response in in vertical motion and
through vertical motion we can say right
where is low pressure likely to develop
where is high pressure likely to
develop.
Although you know you ask me can you
predict the weather using using this
equation because this is a diagnostic
equation it doesn't tell you anything
about how the velocity field is changing
in time then strictly speaking you can't
however you can when we look at
subjective assessment of this use it to
make inferences about where areas of low
pressure are going to develop where
we're going to see an increase in shower
activity where high pressure is likely
to develop and so just looking at this
response term here. So this is in the
vertical velocity. Unfortunately it's
you know this is a very complicated
equation and as humans we still can't
quite look at this and look at the maps
and immediately map one onto the other.
Um we have to make some further
simplification. So one simplification we
make this is this tells you the
threedimensional curvature of omega. Now
you can do some reasoning and um you
know you could say that omega is varying
sinosoidally um as a function of
pressure. So basically what that means
is you could express omega as sin p and
if you differentiate this twice you end
up with d2 omega by dp^ 2 equals minus
sin p which is minus omega. So we can
make similar arguments to this side of
the equation and we can basically say
that all of this is proportional to
minus omega. And if you remember minus
omega from our um when we looked at the
system of coordinates minus omega
corresponds to ascent. So we've actually
by assuming some kind of wavelike
distribution of the vertical velocity in
the horizontal and the vertical we can
make a good assumption that this
response is broadly equivalent to
ascent. So what the forcing terms? Well,
we've got the vertical variation of
voltage vection. So what what does that
mean exactly? So want to think about an
atmosphere that's got a trough in it. So
this might be the 300 mibar surface and
this might be the th00and mibar surface.
And this is a trough. We're always
looking for where the troughs are, where
the ridges are, because the troughs and
the ridges in the upper air usually
correspond to where the high pressures
and the low pressures are in the lower
atmosphere. Although there's a they're
not quite colloccated in vertical,
they're displaced one way or the other.
But if the geostrophic wind is blowing,
vecting this area of positive vorticity.
So trough is an area of positive
vorticity in this direction. At some
later time, this trough is going to be
in this position here. Now if we look at
what's happened to the thickness of the
atmosphere and the thickness is just the
distance between the pressure levels and
we call this H bar and the thickness is
a effectively a measure of the
temperature. If the mean temperature of
the air is colder then the thickness
becomes lower. If the mean temperature
is warmer then the thickness becomes
higher. But at this location here at
some time not we've gone from hn to h1
we've gone from a sort of relatively
large thickness to a lower thickness.
Now what does that mean in terms of the
temperature? It means that the
atmosphere here is cooled. However,
we've got no we've got nothing about
temperature invection here that we're
not thinking about this term at the
moment. There's no mechanism by which we
can cool the atmosphere. And if we
didn't cool the atmosphere then either
it wouldn't be in geostrophic or
hydrostatic balance. So how do we cool
the atmosphere? Well, we basically have
some vertical motion. We have a scent.
If you have a parcel of air and it
rises, rising air expands and it cools.
So this process of vertical motion cools
off the atmosphere. And this is what the
omega equation tells you. It tells you
what is the vertical velocity that I
need in order to cool the atmosphere
enough to reduce the thickness enough to
accommodate this vorticity.
And you make you can make a similar
argument for this. You can um introduce
some thermal adction. And that again, if
you have your same pressure levels and
you introduce some warm adction, maximum
warm adction, you would increase your
thickness because you've introduced
warmer air that bulges up the contours
of the pressure levels at height. And
what does this bulge do? Well, it's it's
the inverse of a trough. Basically, it's
a ridge. It has negative vorticity. If
we've got no mechanism to generate
vorticity through vorticity action,
well, how do we generate vorticity? We
diverge. This is the ice skater effect.
So, if you have an ice skater who's
spinning around
and then they pull their arms in,
they'll spin faster. Or if they pull
their arms out, they'll spin slower.
Well, to reduce the vorticity, if the
atmosphere pulls its arms out, the
vorticity will reduce and we'll create a
ridge. And how do we get this atmosphere
to do this? We have vertical motion.
because vertical motion goes up, it hits
the top layer of the atmosphere, it
spreads out and it creates the necessary
vorticity in order to keep the fields in
geostrophic balance. So these are the
forcing terms. How do we use them? Why
do we use them? Back when the QG system
was first developed, this was a really
key tool into identifying exactly where
the vertical motion was. The best way to
get the vertical motion from these
forcing terms is to calculate it with a
computer. That's kind of how they did
it. But as forecasters we we can't do
that. We can make these assumptions and
we can look at the fields. We identify
where the vorticity is. We identify
where the thermal adection is. And if
you've got positive vorticity, if you've
got warmer advction, that corresponds to
negative omega, which is ascent. And so
by looking at a map of vorticity, you
can immediately see where the areas of
vertical motion in ascending limb and
the descending limb are going to be. And
that also comes into its own when the
forecast model which can calculate
vertical velocity. Now you know high
performance supercomputers and really
sophisticated models they can calculate
vertical velocity but they can also be
wrong in the positions of the troughs
and the ridges. They can have errors and
we can look at satellite imagery and we
can diagnose where these errors are and
we can say right well if this trough was
a bit further back what does that mean
for the vertical motion field? we can we
can always calculate it in our head and
make a forecast based um on that. So
although this equation comes from you
know deep meteorological history um it's
part of an equation set that's no longer
used in much simpler times in numerical
models. Um it's still used in practice
on the bench although with a lot of a
lot of assumptions, a lot of
simplifications in order to give
forecasters this conceptual model of the
atmosphere where things are ascending,
where things are descending and
therefore where areas of low pressure,
whereas high pressure are developing.
>> But you don't crack out the equation and
put numbers in it. You use more this
kind of response and forcing situations.
>> Exly. Yeah. We look at where the we look
at where the vorticity is. We look at
where the thermalction is. We diagnose
qualit qualitatively whether it's
ascending or descending because we can't
calculate the magnitudes of these terms
in our head. We can't say anything about
the size of the response. And there
there are occasions where one force in
turn will say ascent and one force in
turn will say descent and then we have
other techniques that we can use to kind
of break the tie. But in very
qualitative terms, yeah, we're looking
at the vorticity inction and the thermal
invection. And it tells us whether we
can expect upward motion and development
of low pressure at the surface, even
development of, you know, if it if it
links in with a strong frontal gradient,
that can lead to development of a potent
of low pressure and a a name storm or
whether we're looking at descending air
and development of high pressure and
more settled weather. Well, you made it
this far, so well done. If you'd like to
hear more from Dan, more of a personal
interview, a bit about his life, his
nickname at school, his first job as a
forecaster, his backyard weather setup,
that's something you don't want to miss.
Then check out the Number File podcast.
It's a great interview and one not to
miss.
>> Also, just have a tiny little change.
Quantifying tiny, but not to blow up to
infinity. That wouldn't make sense
because I've changed something so so
small. Why have I got an entirely
different solution?
>> We've also been taught that butterflies
flapping their wings can cause cyclones.
>> The butterfly effect, it's like a chain
reaction. It's one thing leads to
another leads to another. But in the in
the sense of humming an equation, you
input something into your equation. It's
like a function machine. Input some
initial