We derive the approximate pressure profiles, density profiles, and temperature profiles of an atmosphere, also called barometric formulas. Our variant of a derivation goes beyond the common standard exercise of a thermodynamics lecture, where commonly the discussion of the underlying physical assumptions is missed. We depart from the Navier-Stokes equation and explicitly point our attention on the physical assumptions disregarded elsewhere. We show that the usual assumptions can be relaxed leading to generalized formulas that hold even in the case of horizontal winds. This fundamental physics has some relevance to the current discussions on the climate debate.
Deep Dive into On The Barometric Formulas And Their Derivation From Hydrodynamics and Thermodynamics.
We derive the approximate pressure profiles, density profiles, and temperature profiles of an atmosphere, also called barometric formulas. Our variant of a derivation goes beyond the common standard exercise of a thermodynamics lecture, where commonly the discussion of the underlying physical assumptions is missed. We depart from the Navier-Stokes equation and explicitly point our attention on the physical assumptions disregarded elsewhere. We show that the usual assumptions can be relaxed leading to generalized formulas that hold even in the case of horizontal winds. This fundamental physics has some relevance to the current discussions on the climate debate.
In the following, we derive approximate temperature profiles of an atmosphere, also called adiabatic lapse rates or better barometric formulas. Our variant of a derivation goes beyond the common standard exercise of a thermodynamics lecture, where commonly the discussion of the underlying physical assumptions is missed. We depart from the Navier-Stokes equation and explicitly point our attention on the physical assumptions disregarded elsewhere. By the way, this derivation is a good example on how to apply the magnetohydrodynamic equations regarded as redundant by some of our critics. Furthermore, it explicitly shows that in physics an application of formulas is valid only in a finite space-time region. In addition, we show that the usual assumptions can be relaxed leading to generalized formulas that hold even in the case of horizontal winds.
A brief historical review of the barometric formula is given in Ref. [1]. The reader is also referred to the textbook by Riegel and Bridger on “Fundamentals of Atmospheric Dynamics and Thermodynamics” [2].
2 On the derivation of the barometric formulas
As described in our falsification paper [4,3] the core of a climate model must be a set of equations describing the equations of fluid flow, namely the generalized Navier-Stokes equations. They describe the conservation of momentum and read
where v is the velocity vector field, p the pressure field, Φ the gravitational potential, R the friction tensor, and F ext are the external force densities, which could describe the Coriolis and centrifugal accelerations. Neglecting the friction term and the electromagnetic fields we obtain the Euler equations.
Assumption 1
• We neglect the electromagnetic field terms.
We get the more common version of the Navier-Stokes equations
The left hand side of this equation may be rewritten according to
With the continuity equation for the mass density ∂̺/∂t + ∇ • (̺v) = 0 this term simplifies to
Thus we obtain the well-know form of the Navier-Stokes equations, or, preferring the singular form, the Navier-Stokes equation
where the term -̺ ∇Φ is gravity. If we neglect the viscosity term, we are left with the Euler equation
Assumption 2
• We assume that v(r, t) is independent of r. In sharp distinction to the standard derivation of the barometric formulas, we relax the usual condition v(r, t) ≡ 0 in order to allow non-vanishing velocity fields v(r, t), which are independent of r.
Consequently, the viscosity tensor R and the non-linear term ̺v • ∇v are zero, such that
Remark: If one writes
one could weaken this assumption to potential velocity fields. With these formulas one can derive the Bernoulli equation.
In case of a rotating atmosphere of the Earth the last term F ext of the right hand side of Eq. 7 describes the centrifugal acceleration and the Coriolis acceleration. The latter vanishes for a identically vanishing velocity field.
• We set F ext to zero.
We now have two fields, namely -∇p and -̺ ∇Φ, which will accelerate the volume elements, if they are different fields:
Let us follow the common notation and write for the gravitational field
Assumption 4
• We assume that, as usual, acceleration due to gravity is vertical, i.e. we set
where e z is the unit vector in z-direction and g is constant in space and time. This is the flat earth hypothesis. Furthermore, we neglect the variation of the gravitation field induced by the gravitation fields of the Sun, the Moon, and the planets.
Assumption 5
• We assume that the wind blows only horizontally, i.e.
Eq. 9 now becomes
That is, with the usual assumptions about geometry we would get the hydrostatic equation
without the usual assumption v(r, t) ≡ 0. In standard thermodynamics for a macroscopic volume the pressure p is characterized by one number, not a field. Irreversible thermodynamics is a (classical) field theory and hydrodynamics is a special case.
The equation of state for an ideal gas reads
where v is the volume of one gram, and R = R/(1 Mol), where R is the usual molar gas constant. Using the density ̺ we also may write, respectively,
Assumption 6
• The air of the atmosphere obeys an equation of state of an ideal gas.
With ̺ = p/( RT ) inserted into Eq. 14 we have
If the molecular mass of the gas is greater, then the decrease of pressure with increase of height will be greater as well. For a temperature field T that is constant in space and time this equation can be integrated.
Assumption 7a
• We postulate an isothermal atmosphere.
which may be integrated to
from which, with ̺ = p/( RT ), one obtains the density as a function of height
Thus, with help of these relations and assumptions, we obtain the barometric height formulas in case of an isothermal atmosphere. The lapse rates for the pressure and density, respectively, depend on the molecular mass of the gas, since R = R/(1 Mol).
In what follows, we need three relations for the heat differential form dQ, namely
Assumption 7b
• For the ideal g
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