Thermodynamic Energy Equation

∂T u∂T v∂T w∂T dT ∂t = − ∂x − ∂y − ∂z + dt (1)

where dT/dt is the individual derivative of temperature. This temperature change experienced by the air parcel itself, dT/dt can be viewed as a "source/sink" term. The first law of thermodynamics allows us to quantify € this "source" term. The term ∂T/∂z is the environmental lapse rate, Γ.

The time derivative of the first law of thermodynamics is

dT g 1 dq = − w + (2) dt c p c p dt

where q is sensible heating or cooling. For adiabatic processes, the far right o -1 hand term is 0. The term – g/cp is 1 C 100 m and is the dry adiabatic lapse rate Γ. € Substitute (2) into (1)

g dq ∂T = −u∂T − v∂T − w∂T − w + 1 (3) ∂t ∂x ∂y ∂z c p c p dt

The definition of the adiabatic and environmental lapse rates:

€ g dT ∂T Γd = = − ,Γ = − (4) c p dz ∂z

can now be substituted into (3). This substitution gives the Temperature Tendency Equation

€ ∂T ∂T ∂T 1 dq = −u − v − w(Γd − Γ) + (5) ∂t ∂x ∂y c p dt

€

Notice that the third right hand term replaces the vertical temperature advection with a term that modifies that vertical temperature advection by adiabatic warming or cooling.

Substitution of the hydrostatic equation into (3) gives the Thermodynamic Energy Equation

% ( dq ∂T = −u∂T − v∂T +ω' dT −∂T * + 1 (6) ∂t ∂x ∂y & dp ∂p) c p dt

as is given as equation (4.3.4) in Bluestein except with the term a/cp for dT/dp. € Static Stability Parameter

The relationship between the modified vertical temperature advection in the rectangular (x,y,z) and isobaric (z, y, p) can be obtained by etting (5) equal to (6) gives:

# dT ∂T & −w(Γd − Γ) = ω% − ( (7a) $ dp ∂ p '

Remember that the relation between omega and vertical velocity is

� ≈ −��

which can be inserted into (7a) to give

$ dT ∂T ' w(Γd − Γ) ≈ −ρgw& − ) (7b) % dp ∂ p (

Rearranging terms gives the static stability parameter

2 (Γd − Γ) # dT ∂T & = % − ( = σ (8) ρg $ dp ∂ p '

For a stable atmosphere, the dry adiabatic lapse rate always exceeds the environmental lapse rate, and the static stability parameter is > 0.

By substitution of (8), Equation (6) may now be rewritten

∂T = −u∂T − v∂T +ωσ + 1 dq (9) ∂t ∂x ∂y cp dt

Brunt-Vaisala Frequency

In ERTH 260, we discussed various ways of visualizing hydrostatic stability. In that series of discussions, we derived Poisson’s Relation, which is

R "1000 % C p Θ = T $ ' (10) # p &

Taking the natural log of both sides gives

ln � = ln � + ln 1000 − ln � (11a,b)

�ln � = � ln � + � ln 1000 − ln �

The partial derivative with respect to height of (11b) is

3 ∂ lnΘ ∂ lnT ∂ ln1000 ∂ ln p C ( ) = C ( ) + R ( ) − R ( ) (12) p ∂z p ∂z ∂z ∂z

Remembering that

� ln � = = (13)

and substituting the gas law and the hydrostatic equation gives

Cp ∂Θ Cp ∂T R ∂p = − Θ ∂z T ∂z p ∂z T ∂Θ ∂T R T ∂p (14a,b) = − Θ ∂z ∂z ρ Cp ∂z

Rearranging terms and using the definition for the lapse rates gives

€ T ∂Θ ∂T g = − = (−Γ + Γd ) = (Γd − Γ) Θ ∂z ∂z Cp (15a,b) T ∂Θ Γd − Γ − = ( ) = σ Θ ∂ p ρg

(15a, b) allow us to characterize stability on the basis of the vertical variation of potential temperature. Equation (15b) states that the static stability is greatest in situations in which the vertical gradient of isentropes is the greatest, that is to say, situations in which there is a large change in potential temperature with height. This will manifest as a packing of the isentropes in cross sections, with the greatest packing in regions in which the atmosphere is MOST stable.1 Thus, isentropes are packed in frontal zones,

1 Remember that the atmosphere is mostly stable. Only in restrictive circumstances is the atmosphere absolutely unstable. In those cases potential temperatures DECREASE with height, and a great packing of isentropes in that circumstance is directly proportional to CAPE.

4 inversions and in the stratosphere, whereas, they are not in regions in which the atmosphere tends toward low static stability.

For a stable atmosphere, the static stability parameter is always positive. In the restrictive and rare case of absolutely unstable conditions, the parameter is negative. For a positive static stability parameter, parcels displaced from an initial elevation will be colder and denser than their surroundings at a given elevation. If the parcel is displaced and released, it will oscillate around its initial elevation until it comes to rest. The period (or frequency) of these oscillations can be appears in the double integration of the vertical equation of motion and obtaining a solution for the height, z.

dw 1 ∂ p = −g − (16) dt ρ ∂ z

Using the gas law and the definition of potential temperature d 2z g ∂Θ = − (17) dt 2 Θ ∂ z

Notice that the right hand side of equation (17) contains a factor proportional to the static stability parameter, asi n (15a).

The solution is in the form of an exponential function, with a power that contains the vertical derivative of potential temperature, as in equations (14a and 15a).

(18)

where z’ is the lower elevation bound of the oscillation, z’o is the height to which the air parcel is force lifted and N is known as the Brunt-Vaisala frequency, given by the expression:

g ∂Θ N = (19) Θ ∂z

Larger values of N occur for highly stable atmospheres and vice versa. N often appears in equations involving vertical instabilities in the atmosphere.

5 The student should keep in mind that the Brunt-Vaisala Frequency is simply another measure of the static stability.

Equation (19) conceptually says that the more stable the atmosphere the higher the amplitude and frequency of the oscillations (the shorter the wavelength). The quantity z’ – z0’ is a way of quantifying the amplitude of the oscillation. If z0 brings the air parcel to a state in which it is warmer than its surroundings, the air parcel has “overshot” its equilibrium level and will oscillate back upwards again until the oscillations are damped.

The hydraulic jump is connected to this physics, but with additional factors, summarized in a different handout.