atmospheric science 4320 / 7320

Atmospheric Science 4320 /

Anthony R. Lupo

Day one Let’s talk about fundamental

Kinematic Concepts

In lab, we talked about divergence, which is a scalar quantity:

Day one We can prove that divergence is the

fractional change with time of some horizontal area A.

yxdtAd

Day one/two Then

finally:

hh VdtdA

Day two We could extend the concept to 3-D and

Day two Horizontal divergence in terms of a

line integral, invoking Green’s theorem (2-D) and Gauss’ (3-D) theorem

Define area A (with tangential wind vectors)

Day two We must also assume Green’s theorem holds

defining a line integral:

FdxdyFdAdsnF ˆ

Day two Green’s theorem (where P = u and Q = v)

S is the oriented surface, or the position vector on the curve is R

thus ds = dr

F is a vector field in the normal direction on S, in our case V (which is tangent to the curve) where we consider:

Day two So F is:

The normal component is:

hn VkV

Day two And we invoke Green’s theorem;

Now recall vector identity: Ax(BxC) = (AdotC)B - C(AdotB)

dAVkdrnV

FdAdsnF

Day two And see that:

And what is the second term on the RHS equal to????

dAkVdAVkdAVkA

hh ˆˆˆ

hh VdAVk

Day two In 3-D we invoke Gauss’s theorem:

Stoke’s Theorem

VdVFFdsdsF

Day two Recall Green’s theorem:

kjNiMyxF ˆ0ˆˆ),(

NNdyMdx

Day two Then

Day two And;

dxxgxMxgxMMdx

dxxgxMdxxgxMMdx

dxyxMdxyxMMdx

Day two And if you don’t understand this:

You might have a ….little …trouble…

Day two Horizontal divergence in Natural

Coordinates:

(s,n,z,t)

The Velocity in Natural Components:

sVV hh ˆ

ssh ˆˆ

Day two so, the horizontal divergence is:

aha, product rule! (which terms will drop

out as 0?

sVV hhhh ˆ

V hhhh ˆˆ

ˆˆˆˆ

Day two Recall:

ˆˆˆ1

Day two Sooo,

Day two In the Above relationship

Term A refers to the speed divergence:

Term B is the directional divergence (Diffluence)

Day two Speed Div

Day two Diffluence (directional div)

Day two In a typical synoptic situation, these terms

tend to act opposite each other:

Confluence and speed increasing:

Diffluence and speed decreasing:

Day two Alternative derivation of horizontal

divergence in natural coordinates:

Then take derivative (product rule again:

cossin

sincos

Day two/three If we “rotate” i and j (x and y) to coincide

with s and n (s and n) then:

1cos,0sin,0 o

Day two/three and,

Then, the celebrated result!

Day three We can also perform our simple-minded

area analysis along the same lines:

nsdtAd

nsdnsdt

Day three Then

finally:

hhhh V

Day three Divergence in the Large-Scale

Meteorological Coordinate System:

The divergence refers to the cartesian coordinate which is an invariant coordinate.

On large-scales we need to take into account the curvature of the earth’s surface.

Day three However, the earth is curved, thus all else

being equal, if we move a large airmass (approximated as 2 - D)northward (southward), the area gets smaller (larger) implying convergence (divergence).

We can consider the parcel moving

upward or downward also (the area or volume):

Day three These discrepencies arise from the fact

that the Earth is a sphere, and thus we cannot hold i, j, and k constant.

Recall we re-worked the Navier - Stokes equations to be valid on a spherical surface:

Day three The equation

1ˆˆˆˆtan

ˆˆtanˆˆˆ

Day three We can also define divergence:

Thus, for example (can you do the rest?)

kwjviukz

V ˆˆˆˆˆˆ33

kwjviu ˆˆ

ˆˆˆˆ

Day three then;

Recall, we defined expressions for:

.....ˆˆ

33 ixk

uvdtjd

ˆˆˆˆˆ

ˆˆtanˆˆˆ

ˆˆtanˆˆ

Day three thus, we get, for the divergence:

2tan33

Day three How important are these “correction terms” for

each scale?

Phenomena Scale Observed

Planetary () 6,000 – 30,000 km

10-6 s-1 - 10-8 s-1

Synoptic (L) 1,000 – 6,000 km

10-5 s-1

Mesoscale 10 – 1,000 km 10-4 s-1 - 10-3 s-1

Day three And;

Then (this approximation is fine too!),

86 102

,10tan

2tan33

Day three Orders of magnitude of Horizontal divergence and

vertical motions:

Scale Divergence (s-1)

w (cm s-1) b s-1

Planetary 10-6 0.5 0.5

Synoptic 10-5 5 5

Mesoscale 10-4 50 50

Microscale 10-3 500 500

Day three/four Vorticity and Circulation/ unit area

Vorticity is a vector whose magnitude is directly proportional to the circulation/unit area of a plane normal to the vorticity vector.

Vorticity = Curl(V), ROT(V), or

Day four We are primarily interested in the vertical

component of vorticity due to circulations in the horizontal plane:

(xi) (eta) (zeta)

ˆˆˆ

ˆˆˆ33

Day four The vertical component:

This is called relative vorticity:

k ˆˆ

Day four Circulation (Kelvin’s Theorem):

consider a closed curve S, and by definition:

R = the position vector dr = change in the position vector

isi sVdsVC

drVC h

Day four from Green’s Theorem:

FdxdyFdAdsnF ˆ

hhh dxdyVdAVdrVC

Day four Thus, circulation per unit area:

We need to show that:

Recall, line integral definition:

A 0lim

CrVdrVC h

Day four Then,

yvxyyu

pathpathpathpathC

Day four And theeen,

reducing

yvxyyu

xuyxxv

Day four So,

Areancirculatio

Day four Thus, we get the vertical component of

Vorticity:

Let’s Examine the vorticity on a sphere:

Vorticity:

kwjviukz

V ˆˆˆˆˆˆ33

Day four Look at using ‘foil’, u-comp only:

kuikzu

juijyu

iuiixu

ˆˆˆˆ

ˆˆˆˆˆˆ

Day four recall cross product rules:

..,ˆˆˆ,ˆˆˆ,ˆˆˆ,0ˆˆ etcjikjkikjiii

uvdtjd

ˆˆˆˆˆ

ˆˆtanˆˆˆ

ˆˆtanˆˆ

Day four Then Vorticity reduces to:

ˆˆˆ

ˆˆˆˆ

Day four but:

,ˆ1ˆ

ˆ,0ˆ

ˆ,ˆ1ˆtanˆ

ˆˆˆˆ

ˆˆˆˆˆtan33

Day four/five Now each component:

tanˆ33

Day five Compare orders of magnitude of the

curvature term:

xv tan

55 10,10

610tan

Day five Now look at orders of magnitude of the

other components:

1610 sr

Day five And

Relative vorticity in the Natural coordinate system:

Recall:

1310,,

A 0lim

Day five And recall

FdAdsF

hhsh RRR

Day five Then,

hsshshsh

VRRVRVRVC

Day five Since:

sss RRRsA

Day five Then apply:

and get :

(A) (B)

Day five Term B, Shear Vorticity (Thanks to C.

Doswell)

Day five Term A; Curvature Vorticity (Thanks C.

Doswell)

Day five Thus, vorticity has two terms (B) shear

term, and (A) curvature term. (e.g. Bell and Keyser, 1993 (Jan MWR).

Thus each can exist independent of the other or together.

Shearing and curvature Vorticity can cancel out even within a rotating fluid!!

Day five In a vortex:

If the air speed increases as we go toward the center such that shear vorticity cancels curvature.

curvshs

Day five Circulation can be irrotational!! This is the

Rankine Vortex model.

An Alternative Vorticity Derviation

Recall:

sVVVVk hhhh ˆ,ˆ33

Day five Then:

V hhh ˆˆˆ

ˆˆˆˆ

Day five Then,

ssRs s ˆˆ,

ˆ1ˆˆ

ˆˆˆ

Day five So then,

sVV hh

shhh ˆˆˆˆ

ˆˆˆ

Day five Absolute Vorticity:

Is the sum of relative vorticity + Planetary vorticity where,

Planetary vorticity:

Day five Continued

Angular acceleration

Day five And

Thus, the vorticity associated with solid rotation is a vector in the direction of the axis of rotation with a magnitude of twice the angular velocity.

Day five The earth’s vorticity vector:

Absolute vorticity:

kfkkpˆˆsin22ˆ

Day five Geostrophic relative vorticity:

vVV gg

Day five If f = fo:

If f varies, then:

Day five/Six The Vorticity Equation and vorticity

Theorems (Bluestein, p 242 – 271)

We can calculate from three approaches:

del cross 3-d equation of motion (vector math)

obtain from circulation theorems Brute force

Day six Derive vorticity equation:

Vorticity:

Equations of motion, how do we get?

Day six After differentiation (How do we do this)? This will give us equation 3) and 4):

Day sixs

Day six Then subtract equation 3) from 4) to get:

Day six Use some old friends:

Day six We get:

rrrrrr

Day six Combine:

Day six We get (using vector and Jacobean

notation) (x,y,z,t) coords!:

This is the traditional Vorticity equation.

Day six But we have the vorticity equation, w/ five

terms! (A) (B) (C)

(D) (E)

kVVt hhaaa

Day six Q: What does each term represent?

Which are sources and sinks? Which strictly move Vorticity around?

A1: Vorticity Advection, Divergence, Tilting, Solenoidal (baroclinic), and Friction.

A2: D,E A3 The rest of ‘em!

Day six/seven The vorticity advection term (A) :

CVA (PVA) AVA (NVA)

Day six/seven A map (term A) (Thanks Dr. Martin!):

Day seven The divergence term (B)

.,,, dect

.,,, incrt

Day seven Obviously, the first two cases are more

prevalent!

.,,, incrt

.,,, dect

Day seven The tilting term (C) (From C. Doswell)

Day seven The solenoidal term (D):

Generation:

Then the left hand side gets a positive

contribution.

Day seven Destruction:

Then the left hand side gets a negative contribution!

Day seven Frictional term (E) :

Increase friction, increase convergence, and the LHS decreases!

Decrease friction, decrease convergence, and the LHS increases!

Day seven The typical orders of magnitude of each term:

Term Synoptic Scale Planetary Scale

1 / s2 1 / s2

Vorticity Advection 1 x 10-9 1 x 10-10

Divergence 1 x 10-10 1 x 10-11

Solenoidal 1 x 10-10 1 x 10-11

Tilting 1 x 10-11 1 x 10-12

Frictional 1 x 10-11 1 x 10-12

Day seven In the free atmosphere, we can

approximate as:

kVVt hhaaa

Day seven The vorticity equation in isobaric

coords

Ffuydt

Ffvxdt

!,2,1 subtractthenx

Day seven We get:

(A) (B) (C) (D)

Note! there is no Solenoidal term here!

Vt aaa

Day seven Vorticity equation in (x,y,,t)

coordinates

Then the (Frictionless) Equation of motion is;

fV a ,

hh VkfM

Day seven Where

get a vorticity equation that includes a tilting term,

but if motion is adiabatic, the no vertical motions

gzTcM p

t haaa ˆ

haaa VV

Day seven Vorticity equation in - coordinates;

Thus, there is a solenoidal term!

t haaa

Day eight Vorticity Theorems: Manipulation and

approximations of the basic vorticity equations.

Constant Absolute Vorticity Trajectories (CAVT).

Consider the isobaric vorticity equation:

Day eight Assume invicid, and no gradient in the

vertical motion:

Assume we are at the non-divergence level:

Thus, we reduce to the “barotropic” vorticity equation!

Day eight (Obviously, vorticity is constant following

lines of absolute vorticity)

Q: Which term survives? What can we do with Vorticity?

A: Term A (advection) and we can move it around!

tconsfr tan

Day eight Assume relative vorticity is all in the curvature

Thus, along the trajectory, with a southerly comp, f is increasing, thus vorticity is decreasing (becoming more anticyclonic),

Then, after some time, we get a northerly component, f is decreasing thus vorticity must increase (becoming more cyclonic).

constfR

Day eight Conservation of Potential Vorticity in

a horizontal flow (quasi- geostrophic PV).

Recall earlier, we derived from the 3-D vorticity equation in x,y,z,t:

Day eight The vorticity equation for horizontal flow

(x,y,,t):

and rewrite:

Day eight We can substitute for the divergence term

by substituting 2-D continuity equation for divergence term:

Couple them:

a lnln

Day eight Thus we can solve to show that:

Day eight So…..

Thus, we get a 2-D PV (Quasi-geostrophic PV) that is conserved following a constant absolute vorticity trajectory (CAVT).

This relationship can show important concepts w/r/t PV, even though this relationship is antiquated and is not used (in favor of IPV or EPV).

constp

Day eight This shows that if:

This relationship can teach us about lee troughing:

gincreagdecreap

gdecreagincreap

elveryadiabaticp

sin,sin0

sin,sin

Day eight Consider a North-South mountain chain

and assume adiabatic:

Assume that d is constant. Then as air climbs mountain, if dp decreases, so must the absolute vorticity (become anticyclonic).

constdp

Day eight Now, f is decreasing, and as we cross the

top of the barrier, dp increases, thus to retain constant values of vorticity, we must get a sharp increase in relative vorticity.

set up a wavelike disturbance that evanesces along the CAVT.

Day eight Flow through an absolute vorticity

maximum, or advection of cyclonic vorticity

We can use the vorticity equation to deduce these patterns.

We know that upper air flow moves much faster than (or flows through) upper air vorticity maxima and upper air waves.

Day six/seven A map (Thanks Dr. Martin!):

Day eight Let’s consider the isobaric form of the

vorticity equation (we’ll only retain the divergence term).

Assume is small or non-existent (steady state vortex, or coherent structure)

hhaa V

hhaah VV

Day eight Thus Divergence will be associated with

Cyclonic Vorticity Advection (where higher values of vorticity are carried toward initially lower values of vorticity)

Convergence will be associated with Anticyclonic Vorticity advection (where lower values of vorticity are carried toward initially higher values of vorticity).

Day eight Again divergence aloft will be maximized

and provide maximum upper level support when it is superimposed over low-level convergence, inducing strong upward motions.

Day nine Divergence / Convergence and

secondary circulations associated with jet streaks: (Uccellini et al, 1984, 1985, 1987, and Uccellini and Kocin, 1987, all MWR)

Let’s first consider the straight line jet:

Divergence/convergence patterns, why there?

Day nine Thus, quadrant by quadrant, div CVA / Conv --

Entrance Exit speed divergence speed

convergence directional conv. directional div. Geostrophic adj. Geostrophic adj. PGF > CO PGF < CO ageo. Comp. plwrd ageo. Comp. eqtrwd.

Day nine Secondary circulations induced by

jet/streaks:

Day nine Q-G perspective

Day nine Consider cyclonically and anticyclonically

curved jets: Keyser and Bell, 1993, MWR.

Day nine Superimpose high and low level jets.

Day nine A bit of lightness…

Day nine Inertial Instability (Hydrodynamic

Instability)

The inertial Stability Criteria of Zonal Geostrophic Flow:

Consider a steady zonal geostrophic current, which varies only in the y - direction. ug = ug(y).

Day nine The horizontal frictionless eq. of motion:

Day nine thus, there is no variation of pressure in x:

and the equations become:

uufdtdv

fvdtdu

Day nine We want to determine if a parcel displaced

northward or southward will be forced back to or accelerate away from the original position. (will north-south KE change w/time)?

Day nine Thus, we want KE, or more appropriately

the sign of KE:

stableV

unstableV

Day nine Displaced parcel (will experience some

change in speed):

thus substituting (from eqns of motion):

dtdtdu

uduuu ggo

fvdtuu go

Day nine at this same point ug is (recall is a function

of y):

vdtdydtdy

Day nine but, the second EOM, the v equation, let’s

subtract,

uffvdt

fvdtuvdty

Day nine multiply by v:

ufdtfv

Day nine In the NH f > 0, v2 > 0, dt > 0

Q: What’s different about the SH?

A: f < 0

Day nine Thus, the stability criteria depends on:

Unstable

Neutral

Stable

Day nine Interpretation:

If the increase in ug to the north (anticyclonic shear), is greater than f (instability - disturbances amplify and reduce the shear to the value of f)

There are no limits in terms of cyclonic shear (stable).

Day nine is rarely but sometimes

observed, usually equatorward of the jet axis (the anticyclonic side). (Why?)

Another way to interpret (vorticity):

Day nine Here:

stable

neutral

unstable

Day nine Interpret in terms of absolute

vorticity (Barotropic instability)

Barotropic instability is important in the terrestrial tropical atmosphere, but is also important in other atmospheres (e.g., Jupiter, Saturn, etc.).

Day nine The absolute vorticity gradient must be

negative for barotropic instability! (e.g., Rayliegh, 1880; Kuo, 1949, JAM, Mudrick, 1974, JAS)

Day nine Then:

Then Barotropic instability depends on the curvature of the geostrophic wind!

Barotropic instability should be favored in “narrow, pointy” jets.

Day ten Introduction to Q-G Theory:

Recall what we mean by a geostrophic system:

2-D system, no divergence or vertical motion no variation in f incompressible flow steady state barotropic (constant wind profile)

Day ten We once again start with our fundamental

equations of geophysical hydrodynamics:

(4 ind. variables, seven dependent variables, 7 equations)

x,y,z,t u,v,w or ,q,p,T or ,

kssourcesdtdm

cQRTP p

Day ten More…….

Day ten Our observation network is in (x,y,p,t).

We’ll ignore curvature of earth:

Our first basic assumption: We are working in a dry adiabatic atmopshere, thus no Eq. of water mass cont. Also, we assume that g, Rd, Cp are constants. We assume Po = a refernce level (1000 hPa), and atmosphere is hydrostatically balanced.

Day ten Eqns become:

lnlnln

Day ten Now to solve these equations, we need to

specify the initial state and boundary conditions to solve. This represents a closed set of equations, ie the set of equations is solvable, and given the above we can solve for all future states of the system.

Thus, as V. Bjerknes (1903) realizes, weather forecasting becomes an initial value problem.

Day ten These (non-linear partial differential equations)

equations should yield all future states of the system provided the proper initial and boundary conditions.

However, as we know, the solutions of these equations are sensitive to the initial cond. (solutions are chaotic).

Thus, there are no obvious analytical solutions, unless we make some gross simplifications.

Day ten So we solve these using numerical techniques.

One of the largest problems: inherent uncertainty in specifying (measuring) the true state of the atmosphere, given the observation network. This is especially true of the wind data.

So our goal is to come up with a system that is somewhere between the full equations and pure geostrophic flow.

Day ten We can start by scaling the terms:

1) f = fo = 10-4 s-1 (except where it appears in a differential)

2) We will allow for small divergences, and small vertical, and ageostrophic motions. Roughly 1 b/s

3) We will assume that are small in the du/dt and dv/dt terms of the equations of motion.

Day ten 4) Thus, assume the flow is still 2 - D.

5) We assume synoptic motions are fairly weak (u = v = 10 m/s).

Also, flow heavily influenced by CO thus ( <<< f).

Day ten 7) Replace winds (u,v,) by their

geostrophic values

8) Assume a Frictionless AND adiabatic atmosphere.

Day ten/11 The Equations of motion and

continutity

So, we could show that the RHS, will nearly cancel each other in these three equations.

Day 11 TIME OUT!

Still have the problem that we need to use height data (measured to 2% uncertainty), and wind data (5-10% uncertainty). Thus we still have a problem!

Much of the development of modern meteorology was built on Q-G theory. (In some places it’s still used heavily). Q-G theory was developed to simplify and get around the problems of the Equations of motion.

Day 11 Why is QG theory important?

1) It’s a practical approach we eliminate the use of wind data, and use more “accurate’ height data. Thus we need to calculate geopotential for ug and vg. Use these simpler equations in place of Primitive equations.

Day 11 2) Use QG theory to balance and replace

initial wind data (PGF = CO) using geostrophic values. Thus, understanding and using QG theory (a simpler problem) will lead to an understanding of fundamental physical process, and in the case of forecasts identifying mechanisms that aren’t well understood.

Day 11 3) QG theory provides us with a

reasonable conceptual framework for understanding the behavior of synoptic scale, mid-latitude features. PE equations may me too complex, and pure geostrophy too simple. QG dynamics retains the presence of convergence divergence patterns and vertical motions (secondary circulations), which are all important for the understanding of mid-latitude dynamics.

Day 11 So Remember……

“P-S-R”

Day 11 Informal Scale analysis derivation of

the Quasi - Geostrophic Equations (QG’s)

We’ll work with geopotential (gz):

Rewrite (back to) equations of motion:(We’ll reduce these for now!)

Day 11 Here they are;

Then, let’s reformulate the thermodynamic equation:

lnlnln

Day 11 Then we have:

on p-sfc, of course:

which we can re-write as:

TccTc 231 ,

Day 11 Then multiply by Specifc Volume and carry

out derivative in two LHS terms:

Then from hydrostatic balance

Day 11 Define static stability (in Q.G. Theory, = (p)):

Thus, we have reworked the first law of thermodynamics. Then applying Q-G theory:

Day 11 Next, let’s rework the vorticity

equation:

In isobaric coordinates:

Day 11 Let’s start applying some of the approximations:

1) Vh = Vg

2) Vorticity is it’s geostrophic value 3) assume zeta is much smaller than f = fo

except where differentiable. 4) Neglect vertical advection 5) neglect tilting term 6) Invicid flow

Day 11 Introduce each assumption:

And we get the vorticity equation. We can substitute for r:

hhoaga

hhaaga

Day 11 to get (recall on LHS ):

and then multiply by fo:

Thus, we are left with two equations in an adiabatic, invicid, Q-G framework.

Day 11 Now let’s derive the height tendency

equation from this set

We will get another “Sutcliffe-type” equation, like the Z-O equation, the omega equation, the vorticity equation.

Like the others before them, they seek to describe height tendency, as a function of dynamic and thermodynamic forcing!

Day 11 Take the thermodynamic equation and:

1) Introduce:

2) switch: 3) apply:

Day 11 And get:

Day 11 Now add the Q-G vorticity and

thermodynamic equation (where ) and we don’t have to manipulate it:

Day 11/12 The result

becomes after addition:

(Dynamics – Vorticity eqn, vort adv)

(Thermodynamics – 1st Law, temp adv)

Day 11/12 This is the original height tendency

equation!

Day 12 The Omega Equation (Q-G Form)

We could derive this equation by taking of the thermodynamic equation, and of the vorticity equation (similar to the original derivation). However, let’s just apply our assumptions to the full omega equation, which we have derived already. (I’ll expect on test)

Day 12 The full omega equation (The Beast!):

Day 12 Apply our Q-G assumptions (round 1):

Assume:

Vh = Vgeo, = g, and r <<< fo f= fo, except where differentiable frictionless, adiabatic = (p) = const.

Day 12 Here we go:

ghaagho

aoghoo

Day 12 Then let’s assume:

1) vertical derivatives times omega are small, or vertical derivatives of omega, or horizontal gradients of omega are small.

2) substitute: 21

Day 12 3) Use hydrostatic balance in temp

advection term.

4) divide through by sigma (oops equation too big, next page)

Day 12 Here we go;

oghooo

Day 12 Of course there are dynamics and

thermodynamics there, can you pick them out?

Q-G form of the Z-O equation (Zwack and Okossi, 1986, Vasilj and Smith, 1997, Lupo and Bosart, 1999)

We will not derive this, we’ll just start with full version and give final version. Good test question on you getting there!

Day 12 Full version:

pt hag

Day 12 Q-G version #1 (From Lupo and Bosart,

1999):

Day 12 Q-G Form #2 (Zwack and Okossi, 1986;

and others)

Day 12 Q-G Form #3!

Day 12 Quasi - Geostropic potential Vorticity

We can start with the Q-G height tendency, with no assumption that static stability is not constant.

Day 12 Multiply by 1/fo:

111 22

Day 12 consider the temperature (thickness)

advection term:

Day 12 we get two terms (from the product rule!).

But, the second term goes to zero! Why?

Day 12 And:

(A) (B)

Since:

TermBTermA

Day 12 So, the equation becomes:

(1) (2)

111 22

Day 12 or terms 2 and 3 can mesh: (1)

or (term 1 – eulerian, term 2 advective!) (hey, I slipped an “f” in there! Why OK?):

Day 12

Vorticity Stability

This is quasi-geostropic potential vorticity! (See Hakim, 1995, 1996, MWR; Henderson, 1999, MWR, March)

Note that we combined dynamic and thermodynamic forcing!!

Day 13 So,

Also, you could start from our EPV expression from earlier this year:

fQGPV o

Day 13 Or in (x,y,p,t) coordinates:

In two dimensions:

constp

Day 13 We assume that:

Thus (recall, this was an “ln” form, so we need to multiply by 1/PV):

wheredtPVd

1 dtPVd

Day 13 so,

Day 13 And “QG”

Day 13 we get QGPV!

Again, we have both thermodynamic and dynamic forcing tied up in one variable QGPV (as was the case for EPV)!

Day 13 Thus, QGPV can also be tied to one

variable, the height field, thus we can invert QGPV field and recover the height field.

We can also “linearize” this equation, dividing the height field into a mean and perturbation height fields, then:

Day 13 Then….

ffQGPVq

QGPVdtd

Day 13 Thus, when we invert the PV fields; we get

the perturbation potential vorticity fields. Ostensibly, we can recover all fields (Temperature, heights, winds, etc. from one variable, Potential Vorticity, subject to the prescribed balance condition (QG)).

Day 13 We’ve boiled down all the physics into one

equation! Impressive development! Thus, we don’t have to worry about non-linear interactions between forcing mechanisms, it’s all there, simple and elegant!

Disadvantage: we cannot isolate individual forcing mechanisms. We must also calculate PV to begin with! Also, does this really give us anything new?

Day 13 Forecasting using QGPV or EPV

Local tendency just equal to the advection (see Lupo and Bosart, 1999; Atallah and Bosart, 2003).

EPVVEPVt

QGPVVQGPVt

EPVdtd

QGPVdtd

Day 13 EPV and QGPV NOT conserved in a

diabatically driven event. Diabatic heating is a source or sink of vorticity or Potential Vorticity.

Potential Vorticity Generation:

.sin FricDiabaticskssourcesEPVdtd

Day 13 Generation:

Day 13/14 The Q - Vector approach (Hoskins et

al., 1978, QJRMS) Bluestein, pp. 350 - 370.

Start w/ “Q-G” Equations of motion:

gyagog

Day 14 Differentiate both w/r/t p:

Day 14 Substitute the thermal wind relationship:

Day 14 And substitute

Day 14 Here is the adiabatic form of the Q-G

thermodynamic equation:

Day 14 Differentiate the 1st Law first w/r/t x then

w/r/t y:

Day 14 Then, multiply Q-G equation of motion

(Eqs. 1 and 2) by:

and add to thermodynamic equations (Eqs. 3 and 4)

Day 14 This gives us Q1 and Q2:

ˆ2ˆ1

Day 14 Then differentiate Q1 and Q2, w/r/t x and

y, respectively (in other words, take divergence).

Day 14 Then use continuity:

This give us the omega equation in Q-vector format!

Day 14 Then since :

This give us a Q-vector with diabatics!

Day 14 Note that on the RHS, we have the

dynamic and thermodynamic forcing combined into one term.

Also, note that we can calculate these on p -surfaces (no vertical derivatives). The forcing function is exact differential (ie, not path dependent), and dynamics or thermodynamics not neglected.

Day 14 This form also gives a clear picture of

omega on a 2-D plot:

Div. Q is sinking motion:

Day 14 Conv. Q is rising motion:

Day 14 Forcing function is “Galilean Invariant”

which simply means that the forcing function is the same in a fixed coordinate system as it is in a moving one (i.e., no explicit advection terms!)

And this is the end of Dynamics! See you next year in 9350?

Day 15 Symmetric Instability

Convective instability:

Measure based on the slope of Γe to Γd or Γe to Γd. Static stability.

It’s an instability in the vertical.

Day 15 Then the slope of the observed potential

temperatures (θ) are important.

If the slope of the dry adiabats is 0, then a “negative” represents stability or “flatter” insentropes. This is a smaller dT/dz.

The opposite is true of dry adiabats which are “rotated” 90 degrees!

Day 15 With Inertial instability, recall:

Stable when absolute vorticity is less than zero: ζa < 0 which was

Could also use:

Day 15 Thus geostrophic momentum is:

(Draw typical N-S):

gyg Mvfx

gxg Mufy

Day 15 Horizonal instability! Then if the lines of

Mg are “flatter”, then neutral. When slanted, parcels can be displace “slantwise”.

So, for “symmetric” instability, we’re going to combine the two situations.

(Draw)

Day 15 Thus, when the Mg lines are sloped more

steeply than θe, then we have symmetric stability. (stable to both vertical and horizontal conditions)

When the θe lines are sloped more than the Mg lines, then we have “symmetric instability”. (unstable for both conditions).

Day 15 Symmetric instability, but the nature of

the presence of temperature gradients, is baroclinic in nature.

General Instability Problem.

We want a statement that might include both dynamic (barotropic) and thermodynamic (convective / baroclinic)instability.

Day15 Start with QGPV:

Then (in QG form):

Day 15 If we only look in the y-z plane

Then take partial / partial-y:

Day 15 And Finally

Barotropic BaroclinicInstability Instability

Day 15

Day 14 Turn out the lights!

Day 14

atmospheric science 4320 / 7320

Documents

*tm 10-4320-344-24p-3

atmospheric science 4320 / 7320

atms 4320 / 7320 lab on the use of isentropic maps in...

lab 6 – atms 4320 / 7320

edit 7320 literature review presentation

· 290tËz500l 150tÆz350l 720 1680 2160 3360 4320 6720...

4320 wireless position monitor

lab 4 – atms 4320 / 7320 the nocturnal jet and severe...

atmospheric science 4320 / 7320 anthony r. lupo. syllabus ...

700487 a lsh-7320 · march 2019 7 lsh-7320 chapter 1:...

7320 electromagnetic theory ii university of colorado

syllabus for fir 4320 / 6320 - m50 (spreal estate finance -...

videojet 4320 operator manual

4320 - clarion manual.pdf · title: 4320.cdr author: vinsam...

im/7320 8 monitors for organic pollution, surrogate color...

tm 5-4320-218-15

deep sea 7320

hoo ,qvslurq ø...support.euro.dell.com...

uninsured has 4320. the quiz http:

aspire 4720g 4720z 4720 4320