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ESCI 343 – Atmospheric Dynamics II Lesson 1 – Q-G Height-tendency Equation
Reference: An Introduction to Dynamic Meteorology (3rd^ edition), J.R. Holton Synoptic-dynamic Meteorology in Midlatitudes, Vol 1 , H.B. Bluestein Reading: Holton, Chapter 6 (Section 6.3)
THE QUASI-GEOSTROPHIC THERMODYNAMIC ENERGY EQUATION
The quasi-geostrophic form of the thermodynamic energy equation in pressure coordinates is
d p
g p c
J
R
p V T t
^ −
Quasi-geostrophic thermodynamic energy equation
where σ is the static stability parameter, defined as
∂ p
θ σ α ln
The static stability parameter is positive for a stable atmosphere, and negative for an unstable atmosphere. Note that for horizontal advection, only the geostrophic wind is used.
From the hydrostatic equation in pressure coordinates we have
p
RT
p
=− = −^ d ∂
Solving for T gives
R p
p T d ∂
Using this expression for temperature in the Q-G thermodynamic energy equation gives
p
R
c
J
p
V
t
d p
g p ∂ + =−
Rearranging terms we get
p
d g p pc
R J
p
V
p t
σω
And finally, if we define the geopotential tendency as
∂ t
then the Q-G thermodynamic energy equation becomes
p
d g p pc
R J
p
V
p
χ ^
. Equation (1)
Keep in mind that equation (1) is really nothing more than the thermodynamic energy equation.
THE Q-G VORTICITY EQUATION REVISITED
The Q-G vorticity equation in pressure coordinates is
p
V f f t g p g
g ∂
The geostrophic vorticity in terms of the geopotential is
= ∇^2 Φ 0
g p f
Substituting this into the Q-G vorticity equation gives another form of the Q-G vorticity equation
p
f f f
f Vg p p ∂
0
0
2 ^1. Equation 2
Equations 1 and 2 are an alternate form of the Q-G system. They are two equations with two dependent variables, χ and ω. If we know the what the geopotential field is, then these equations form a complete system which can be solved for either χ or ω.
THE GEOPOTENTIAL TENDENCY EQUATION
The first equation we will derive is the geopotential tendency equation, found by eliminating ω between Eqns. (1) and (2). The idea behind this is simple, but the individual mathematical steps become complicated. None-the-less, by differentiating Eqn. (1) with respect to pressure, then multiplying it by f 02 /σ, and then adding this to Eqn. (2), a very ugly equation is found (note that for ease of notation the subscript p is no longer written on the del operator, but is implied).
p
f p
J c p
f R p
V p
f f f
fV p
f p
d g g ∂
∂ −
∂
∂ −
∂
∂Φ − •∇ − ∂
∂ −
=− •∇ ∇ Φ+
∂
∂ ∇ + σ ω σ σ σ
χ σ
2 0 2 0 2 2 0 0
2 0
2 2 2 0 ^1
The static stability parameter normally increases with height; however, analysis of
the Q-G tendency equation is slightly easier if we assume that σ is constant so that the
last term disappears. In this case, the equation becomes
∂
∂ −
∂
∂Φ − •∇− ∂
∂ −
=− •∇ ∇Φ+
∂
∂ ∇ + p
J c p
f R p
V p
f f f
fV p
f p
d σ χ g σ g σ
2 0
2 (^20) 0 2 0
2 2 2 0 ^1 .
Q-G geopotential tendency equation
Though ugly, this equation has a sort of inner beauty. We first try to see this beauty by analyzing the terms of the equation in a qualitative fashion. We do this by imaging that the horizontal structure of disturbances in the atmosphere can be approximated by sinusoidal functions such as
χ ( , x y p t , , ) = Χ( p t , ) exp[ ( i k x + l y )].
c. The differential diabatic heating term. The differential heating term (third term on RHS) behaves similarly to the differential thermal advection term. It the heating decreases with height, or cooling increases with height, then heights will rise.
Another useful way of writing the essence of the Q-G tendency equation is
∂∂Φ^ ∂∂ΦΦΦ / ∂∂∂∂ t^ ∝∝∝∝^ −−−− absolute vorticity advection + ∂∂∂∂ / ∂∂∂∂ p (thermal advection) + ∂∂∂∂ / ∂∂∂∂ p (heating) or
∂∂∂∂ΦΦΦΦ / ∂∂∂∂ t ∝∝∝∝ −−−− absolute vorticity advection −−−−∂∂∂∂ / ∂∂∂∂ z (thermal advection) −−−−∂∂∂∂ / ∂∂∂∂ z (heating)
The previous analysis leads us to a very important conclusion. In quasi- geostrophic theory, there are only three ways for heights to fall…either through positive vorticity advection, through warm advection that increases with height, or through diabatic heating that increases with height****!
A PHYSICAL INTERPRETATION OF THE TENDENCY EQUATION
The effects of the terms on the RHS of the tendency equation can be explained physically, as well as mathematically.
a. Vorticity advection : The physical explanation of the vorticity advection term is rather obvious…we know that positive vorticity is associated with low heights, so if vorticity is increased due to advection, the heights will fall. This term cannot amplify the absolute vorticity of the disturbance, but only serves to propagate it.
b. Differential thermal advection : The effects of differential thermal advection can be thought of as follows.:
Imagine a scenario where there is net warm advection in the lower levels (below 500 mb), and net cold advection in the upper levels (above 500 mb). Since the thickness between two pressure surfaces is proportional to temperature, the low-level warm advection will lead to increased thickness of the 1000 – 500 mb layer, while the upper- level cold advection will lead to decreased thickness of the 500 – 200 mb layer. The net result is height rises at 500 mb (see diagram below).
The same result will occur with weak warm advection aloft and stronger warm advection in the low-levels (see diagram below).
If the advection is the same strength both aloft and below, then there is no change in height at 500 mb (see diagram below).
Typically, thermal advection is very small in the upper troposphere (above 500 mb) compared to that in the lower-levels, so it is really the low level advection that determines the 500 mb geopotential tendency. Cold advection in the lower levels will decrease the thickness of the 1000 –500 mb layer, and lower the heights at 500 mb, as would be expected (since cold advection decreasing with height is the same as warm advection increasing with height ). Warm advection in the lower levels will increase the thickness of the 1000 –500 mb layer, and result in height rises at 500 mb.
c. Differential diabatic heating. The physical interpretation for the differential diabatic heating term is similar to that for differential advection. If there is more heating above a level than below it, the heights at that level will fall. Phrased another way, we can say that above the level of maximum heating ( J / p ) the heights will rise, and below the level of maximum heating heights will fall.
Le CHATELIER’S PRINCIPLE
Le Chatelier’s Principle, named for Henry Louis Le Chatelier, states that a thermodynamic system will resist changes in temperature (and other thermodynamic properties), and if forced such that the temperature changes, will react with process that try to restore the original temperature. Though Le Chatelier’s Principle isn’t as rigorous and general as often thought to be^1 , we can see Le Chatelier’s principle at work in the differential thermal advection and diabatic heating terms of the Q-G tendency equation. For example, cold advection (or diabatic cooling) over warm advection (or diabatic heating) forces height rises at 500 mb, as well as height falls at 200 and 1000 mb (as per the diagram below.)
(^1) see J. de Heer, J. Chem. Educ. , 34 , 375 (1957)
If static stability decreases with height, then the effect of vertical motion is opposite from that just described, since there will be larger heating (or cooling) below, rather than aloft.
AN ADVANCED TREATMENT OF THE TENDENCY EQUATION
Since we’ve previously assumed that disturbances in the atmosphere have a sinusoidal horizontal structure, we will also assume that the forcing (terms on the RHS of the tendency equation) also have a sinusoidal structure. So, we assume ( , , , ) ( , ) exp[ ( )] ( ) exp[ ( )] ( ) exp[ ( )]
( ) exp[ ( )]
v T
J
x y p t p t k x l y vorticity advection term F p k x l y dF p thermal advection term k x l y dp dF p diabatic heating term k x l y dp
χ = Χ − + = − +
= − − +
and put these into the tendency equation. This gives an ordinary differential equation,
2 0
2 2 2
2 0
2 2
2
f
k l
dp
dF dp
dF F dp f
d (^) T J v
(we’ve assumed the static stability parameter, σ, is constant with height). The solutions to this equation are hyperbolic sines and cosines with a characteristic vertical length scale of λ=1/Κ. So, the longer the wavelength of a disturbance (lower values of Κ indicate longer wavelengths), the deeper the effects of its forcing terms are felt in the atmosphere.
QUASI-GEOSTROPHIC POTENTIAL VORTICITY
The tendency equation (ignoring the diabatic heating term, J ) can be written as
0
0
^ =
p
f p
f Dt f
Dg
The quantity in brackets is called the quasi-geostrophic potential vorticity ,
p
f p
f f
q σ
(^20) 0
and is conserved following a fluid parcel.
EXERCISES
1. Derive the Q-G tendency equation, showing all steps.
2. Is the vertical extent of the forcing terms in the Q-G tendency equation larger or smaller in the tropics as compared to the middle latitudes?
