# Presence of Diabatic heating term in the continuity equation

The tropical circulation paper by Gill (1980) approximates vertical velocity as the sum of pressure and diabatic heating terms (equation 2.5 at page 449).

Since the pressure is already a function of Q, what's the need of writing Q explicitly in this expression? Doesn't it double count p?

• So 2.5 comes from the buoyancy equation. Do you know what that is? – gansub Jun 17 '19 at 9:23
• Are you talking about the equation containing vertical velocity and pressure perturbation? – Agni Jun 17 '19 at 9:49
• Buoyancy equation here and everywhere else is the same. Look up Brunt Vaisala equation. – gansub Jun 17 '19 at 9:49
• By the way I do not think P is a function of Q. The P term relates to adiabatic contributions and Q diabatic – gansub Jun 17 '19 at 10:52
• I am surprised! Doesn't the amount of diabatic heating in clouds also affect the ambient pressure? The case has to be similar with radiation as well? – Agni Jun 18 '19 at 9:27

In the book - Atmosphere-Ocean Dynamics, Volume 30 the author AE Gill defines the perturbation pressure in the paper that OP references as the following.

Consider a pressure of a reference system (ocean/atmosphere at rest) and let us call it $$p_0$$.

It must be noted that a real atmosphere is never at rest and so we consider deviations from the reference system.

Since we are dealing with a vertical pressure gradient for OP's equation(2.5)(the gravitational force matches the vertical pressure gradient for hydrostatic large scale circulations)

$$p_0$$ is a function of $$z$$ where z is the height coordinate.

Then the perturbation pressure $$p'$$ is defined in the following way

$$p = p_0(z) + p'$$

So if you add the perturbation pressure to the reference pressure we get the deviation from the reference system.

Similarly a perturbation density can be defined in the following way

$$\rho = \rho_0(z) + \rho'$$

The basis for doing this comes from classical mechanics Perturbation Theory

The question is why is this being done ?

We are looking for solutions that are basically of a oscillatory nature. So in effect this gets rid of the non linear cross terms for which no oscillatory solution exists.

In the specific case of 2.5 from the paper you referenced in addition to the Brunt–Väisälä frequency one needs to consider a "buoyancy forcing" especially in the tropics because of the large contributions from diabatic heating. We are essentially talking of an "open system" where in addition to adiabatic "work" being done you also have to account for heat exchange between a system and the ambient environment. In reality this is assumed as "pseudoadiabatic" process

Then the question is how is the heating rate modeled ? In the AE Gill book this is given by the following equation

$$Q_H = -L_V \frac{Dq_w}{Dt}$$

where $$L_v$$ is the latent heat of vaporization and $$q_w$$ is the saturation humidity where the word saturation means the saturation water vapor mixing ratios.

Equation (2.5) from OP's paper is derived in Gill's book and is present in the 9th chapter(9.13.7) and will not be derived here but will be stated as is

$$\omega_n = \frac{\partial \widetilde \eta}{\partial t} + \widetilde b_n$$

Here $$\widetilde \eta$$ is the vertical coordinate(in OP's paper the vertical coordinate is pressure) and $$b_n$$ is the rate of change of buoyancy per unit volume. This form is known as the buoyancy forced Shallow water equations

One can get more details about the perturbation theory for the atmosphere by looking at this link - Atmospheric Oscillations: Linear Perturbation Theory

Similarly if you want horizontal pressure perturbations you model the reference pressure as a function of x and y and then consider deviations thereof.

• It seems to me that the notion of pressure is based upon an approximation which considers the fluid in thermal equilibrium. If I am not wrong, the horizontal momentum equations are also independent of diabatic forcings as the horizontal pressure gradient doesn't deal with it? What's the situation in models then? The solutions might not show sensitivity to diabatic heating then? Sounds like a serious problem!! – Agni Jun 19 '19 at 5:44
• As per my understanding, models include this term only in vertical momentum equation, the horizontal momentum, pressure, density and theta are still independent of it. How reliable are these quantities then? – Agni Jun 19 '19 at 6:22
• @Agni In meteorology and atmospheric science the fluid is not considered to be in thermal equilibrium. Only the reference state is. If you consider the MJO non linear advection can move the system beyond the equatorial waveguide. Convective heat in GCMs is usually parameterized and not considered explicitly. Cloud resolving models may do this explicitly. ecmwf.int/sites/default/files/elibrary/2011/… – gansub Jun 19 '19 at 7:05
• Okay, so the pressure perturbation, density and other quantities must show the dependence upon diabatic heating in CRMs? Do you know which equation is used to evaluate this dependence? I use a CRM but I don't see an explicit dependence through the governing equations, it is also not evident in some of my simulations. – Agni Jun 19 '19 at 7:11
• What will be the benefit of a coupled model if I intend to evaluate the response of cloud induced heating on ambient surroundings? – Agni Jun 19 '19 at 7:19