Barotropic fluid

for a barotropic fluid. It gives the pressure contribution to a steady Bernoulli equation. In isentropic thermodynamics it agrees with specific enthalpy up to a constant; for an externally maintained isothermal ideal gas it is

c_{s}^{2} lo g ρ

and is not the constant thermodynamic enthalpy at fixed temperature.

Centrifugal potential of cylindrical rotation ( $Ψ = Φ - \int_{0}^{R} R^{'} Ω^{2} (R^{'}) d R^{'}$ )

 0  0

For angular velocity depending only on cylindrical radius, the centrifugal acceleration is a gradient. Steady barotropic balance is

\nabla p / ρ = - \nablaΨ

, hence specific enthalpy satisfies

h (ρ) + Ψ = constant

. On an invertible equation-of-state branch, mass density and pressure are functions of

Ψ

. Cylindrical rotation is essential to this scalar centrifugal-potential representation.

Barotropic energy density ( $ρ U^{'} - U = P$ )

 0  0

For a barotropic fluid with pressure

P = P (ρ)

, choose an energy density

U

satisfying

ρ U^{'} (ρ) - U = P

. The continuity equation then gives

\partial_{t} U + \nabla \cdot (U u) = - P \nabla \cdot u

. For an isothermal equation of state

P = c_{s}^{2} ρ

, one choice is

U = c_{s}^{2} ρ ln (ρ / ρ_{ref})

. A fixed reference density changes

U

only by a conserved mass term. The same construction holds with surface density and vertically integrated pressure. This closure energy need not equal the microscopic internal energy of a gas maintained at fixed temperature.

Barotropic magnetic energy equation

 0  0

For a barotropic fluid, an energy per mass

e

can be defined by

d e / d ρ = p / ρ^{2}

. For the isothermal equation of state,

e = c_{s}^{2} ln (ρ / ρ_{0})

. Adding kinetic, gravitational and magnetic energy gives the ideal magnetic energy flux

u (E + p + B^{2} / (2 μ_{0})) - B (u \cdot B) / μ_{0}

. This is a mechanical barotropic energy identity; it does not say that an isothermal gas is thermally isolated.

Barotropic vorticity transport ( $\partial_{t} ω = \nabla \times (u \times ω)$ )

 0  0

For an inviscid barotropic fluid with only potential body forces, the vorticity

ω = \nabla \times u

obeys

\partial_{t} ω = \nabla \times (u \times ω) .

(1)

The pressure force is a gradient because

ρ^{- 1} \nabla p = \nabla \int^{ρ} p^{'} (s) d s / s

. A spatially uniform specific entropy makes a fixed-composition fluid barotropic. Merely having

Ds / D t = 0

does not exclude baroclinic vorticity generation from spatial entropy gradients.

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A barotropic fluid is a type of fluid whose pressure is a function solely of its density. In simpler terms, for a barotropic fluid, if you know the density at a given point, you can determine the pressure without needing to know any other state variables, such as temperature or specific internal energy. In mathematical terms, the relationship can be expressed as: \[ P = P(\rho) \] where $ P $ is the pressure and $ \rho $ is the density.

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