Hamiltonian description of idealized binary geophysical fluids

Peter R. Bannon

Research output: Contribution to journalArticle

15 Citations (Scopus)

Abstract

A Hamiltonian formulation for the dynamics and thermodynamics of a compressible, rotating, binary fluid subject to gravity is developed. Here, binary refers to the presence of two components of the fluid, such as solids dissolved in a liquid or gaseous and liquid water existing along with dry air. These fluids are idealized in that the influences of diffusion processes are ignored and the binary flow is restricted to a single velocity. The equations are presented in generic form applicable to an arbitrary binary geophysical flow. The relevant Poisson bracket satisfies Jacobi's identity. Three distinct Casimir invariants are described. The first reflects the conservation of entropy and concentration of the minor component. The second is a consequence of the conservation of the absolute circulation on curves formed by the intersection of surfaces of constant entropy with surfaces of constant concentration. The third is a generic potential vorticity of the form (ω · ∇λ)/ρ. Here, ω is the absolute vorticity, ρ is the total density of the fluid, and λ is any thermodynamic variable. For example, λ can be the pressure, density, temperature, or mixing ratio as well as the more common choice of potential temperature. Available energy of the system is defined locally in the finite-amplitude as well as in the small-amplitude limit. Both definitions are partitioned into available potential and available elastic energies. A linear stability analysis indicates that the fluid is statically stable provided the square of the sound speed is positive, the total density decreases with height, and the square of a suitably defined buoyancy frequency is positive. The formulation is applicable to a salty ocean and to a moist atmosphere. For the atmosphere, the full theory holds in the presence of either liquid water or ice in equilibrium with its vapor.

Original languageEnglish (US)
Pages (from-to)2809-2819
Number of pages11
JournalJournal of the Atmospheric Sciences
Volume60
Issue number22
DOIs
StatePublished - Nov 15 2003

Fingerprint

fluid
liquid
entropy
thermodynamics
rotating fluid
atmosphere
potential temperature
potential vorticity
stability analysis
vorticity
mixing ratio
buoyancy
energy
gravity
ice
water
air
ocean
temperature
sound

All Science Journal Classification (ASJC) codes

  • Atmospheric Science

Cite this

Bannon, Peter R. / Hamiltonian description of idealized binary geophysical fluids. In: Journal of the Atmospheric Sciences. 2003 ; Vol. 60, No. 22. pp. 2809-2819.
@article{ba842ac9dfc445a0b89ac91316432f3a,
title = "Hamiltonian description of idealized binary geophysical fluids",
abstract = "A Hamiltonian formulation for the dynamics and thermodynamics of a compressible, rotating, binary fluid subject to gravity is developed. Here, binary refers to the presence of two components of the fluid, such as solids dissolved in a liquid or gaseous and liquid water existing along with dry air. These fluids are idealized in that the influences of diffusion processes are ignored and the binary flow is restricted to a single velocity. The equations are presented in generic form applicable to an arbitrary binary geophysical flow. The relevant Poisson bracket satisfies Jacobi's identity. Three distinct Casimir invariants are described. The first reflects the conservation of entropy and concentration of the minor component. The second is a consequence of the conservation of the absolute circulation on curves formed by the intersection of surfaces of constant entropy with surfaces of constant concentration. The third is a generic potential vorticity of the form (ω · ∇λ)/ρ. Here, ω is the absolute vorticity, ρ is the total density of the fluid, and λ is any thermodynamic variable. For example, λ can be the pressure, density, temperature, or mixing ratio as well as the more common choice of potential temperature. Available energy of the system is defined locally in the finite-amplitude as well as in the small-amplitude limit. Both definitions are partitioned into available potential and available elastic energies. A linear stability analysis indicates that the fluid is statically stable provided the square of the sound speed is positive, the total density decreases with height, and the square of a suitably defined buoyancy frequency is positive. The formulation is applicable to a salty ocean and to a moist atmosphere. For the atmosphere, the full theory holds in the presence of either liquid water or ice in equilibrium with its vapor.",
author = "Bannon, {Peter R.}",
year = "2003",
month = "11",
day = "15",
doi = "10.1175/1520-0469(2003)060<2809:HDOIBG>2.0.CO;2",
language = "English (US)",
volume = "60",
pages = "2809--2819",
journal = "Journals of the Atmospheric Sciences",
issn = "0022-4928",
publisher = "American Meteorological Society",
number = "22",

}

Hamiltonian description of idealized binary geophysical fluids. / Bannon, Peter R.

In: Journal of the Atmospheric Sciences, Vol. 60, No. 22, 15.11.2003, p. 2809-2819.

Research output: Contribution to journalArticle

TY - JOUR

T1 - Hamiltonian description of idealized binary geophysical fluids

AU - Bannon, Peter R.

PY - 2003/11/15

Y1 - 2003/11/15

N2 - A Hamiltonian formulation for the dynamics and thermodynamics of a compressible, rotating, binary fluid subject to gravity is developed. Here, binary refers to the presence of two components of the fluid, such as solids dissolved in a liquid or gaseous and liquid water existing along with dry air. These fluids are idealized in that the influences of diffusion processes are ignored and the binary flow is restricted to a single velocity. The equations are presented in generic form applicable to an arbitrary binary geophysical flow. The relevant Poisson bracket satisfies Jacobi's identity. Three distinct Casimir invariants are described. The first reflects the conservation of entropy and concentration of the minor component. The second is a consequence of the conservation of the absolute circulation on curves formed by the intersection of surfaces of constant entropy with surfaces of constant concentration. The third is a generic potential vorticity of the form (ω · ∇λ)/ρ. Here, ω is the absolute vorticity, ρ is the total density of the fluid, and λ is any thermodynamic variable. For example, λ can be the pressure, density, temperature, or mixing ratio as well as the more common choice of potential temperature. Available energy of the system is defined locally in the finite-amplitude as well as in the small-amplitude limit. Both definitions are partitioned into available potential and available elastic energies. A linear stability analysis indicates that the fluid is statically stable provided the square of the sound speed is positive, the total density decreases with height, and the square of a suitably defined buoyancy frequency is positive. The formulation is applicable to a salty ocean and to a moist atmosphere. For the atmosphere, the full theory holds in the presence of either liquid water or ice in equilibrium with its vapor.

AB - A Hamiltonian formulation for the dynamics and thermodynamics of a compressible, rotating, binary fluid subject to gravity is developed. Here, binary refers to the presence of two components of the fluid, such as solids dissolved in a liquid or gaseous and liquid water existing along with dry air. These fluids are idealized in that the influences of diffusion processes are ignored and the binary flow is restricted to a single velocity. The equations are presented in generic form applicable to an arbitrary binary geophysical flow. The relevant Poisson bracket satisfies Jacobi's identity. Three distinct Casimir invariants are described. The first reflects the conservation of entropy and concentration of the minor component. The second is a consequence of the conservation of the absolute circulation on curves formed by the intersection of surfaces of constant entropy with surfaces of constant concentration. The third is a generic potential vorticity of the form (ω · ∇λ)/ρ. Here, ω is the absolute vorticity, ρ is the total density of the fluid, and λ is any thermodynamic variable. For example, λ can be the pressure, density, temperature, or mixing ratio as well as the more common choice of potential temperature. Available energy of the system is defined locally in the finite-amplitude as well as in the small-amplitude limit. Both definitions are partitioned into available potential and available elastic energies. A linear stability analysis indicates that the fluid is statically stable provided the square of the sound speed is positive, the total density decreases with height, and the square of a suitably defined buoyancy frequency is positive. The formulation is applicable to a salty ocean and to a moist atmosphere. For the atmosphere, the full theory holds in the presence of either liquid water or ice in equilibrium with its vapor.

UR - http://www.scopus.com/inward/record.url?scp=0346689005&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0346689005&partnerID=8YFLogxK

U2 - 10.1175/1520-0469(2003)060<2809:HDOIBG>2.0.CO;2

DO - 10.1175/1520-0469(2003)060<2809:HDOIBG>2.0.CO;2

M3 - Article

AN - SCOPUS:0346689005

VL - 60

SP - 2809

EP - 2819

JO - Journals of the Atmospheric Sciences

JF - Journals of the Atmospheric Sciences

SN - 0022-4928

IS - 22

ER -