• Bannon, P., 2002: Theoretical foundations for models of moist convection. J. Atmos. Sci., 59 , 19671982.

  • Das, P., 1979: A non-Archimedean approach to the equations of convection dynamics. J. Atmos. Sci., 36 , 21832190.

  • Davies-Jones, R., 2002: Linear and nonlinear propagation of supercell storms. J. Atmos. Sci., 59 , 31783205.

  • Duff, G. F. D., , and D. Naylor, 1966: Differential Equations of Applied Mathematics. Wiley, 423 pp.

  • Emanuel, K. A., 1994: Atmospheric Convection. Oxford University Press, 580 pp.

  • Glickman, T. S., Ed.,. . 2000: Glossary of Meteorology. 2d ed. Amer. Meteor. Soc., 855 pp.

  • Leslie, L. M., , and R. K. Smith, 1978: The effect of vertical stability on tornadogenesis. J. Atmos. Sci., 35 , 12811288.

  • Markowski, P. M., , J. M. Straka, , and E. N. Rasmussen, 2002: Direct surface thermodynamic observations within the rear-flank downdrafts of nontornadic and tornadic supercells. Mon. Wea. Rev., 130 , 16921721.

    • Search Google Scholar
    • Export Citation
  • Markowski, P. M., , J. M. Straka, , and E. N. Rasmussen, 2003: Tornadogenesis resulting from the transport of circulation by a downdraft: Idealized simulations. J. Atmos. Sci., 60 , 795823.

    • Search Google Scholar
    • Export Citation
  • Soong, S-T., , and Y. Ogura, 1973: A comparison between axisymmetric and slab-symmetric cumulus cloud models. J. Atmos. Sci., 30 , 879893.

    • Search Google Scholar
    • Export Citation
  • Trapp, R. J., , and R. Davies-Jones, 1997: Tornadogenesis with and without a dynamic pipe effect. J. Atmos. Sci., 54 , 113133.

  • Weisman, M. L., , and R. Rotunno, 2000: The use of vertical wind shear versus helicity in interpreting supercell dynamics. J. Atmos. Sci., 57 , 14521472.

    • Search Google Scholar
    • Export Citation
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An Expression for Effective Buoyancy in Surroundings with Horizontal Density Gradients

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  • 1 NOAA/National Severe Storms Laboratory, Norman, Oklahoma
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Abstract

An expression is obtained for effective buoyancy per unit volume in inviscid anelastic flow with surrounding horizontal density variations. Effective buoyancy is defined here as the statically forced part of the locally nonhydrostatic, upward pressure-gradient force because this is the part of the net vertical force that is both absolute (i.e., independent of an arbitrary base state) and dependent solely on variations in the specific weight of the air–hydrometeor mixture. For the case where the domain is the half space z > 0 above flat ground, effective buoyancy β(x) at a given field point x is given by β(x) = −∞ −∞ −∞ [g2HρT()]/(4π|x|) dx̂ dẑ, where T is the total weight of air and hydrometeors per unit volume, the ≡ (x̂, ŷ, ) are source points, 2H is the horizontal Laplacian with respect to , and 2HρT(x̂, ŷ,) ≡ −2HρT(x̂, ŷ, ).

Corresponding author address: Dr. Robert Davies-Jones, NOAA/National Severe Storms Laboratory, 1313 Halley Circle, Norman, OK 73069-8493. Email: Bob.Davies-Jones@noaa.gov

Abstract

An expression is obtained for effective buoyancy per unit volume in inviscid anelastic flow with surrounding horizontal density variations. Effective buoyancy is defined here as the statically forced part of the locally nonhydrostatic, upward pressure-gradient force because this is the part of the net vertical force that is both absolute (i.e., independent of an arbitrary base state) and dependent solely on variations in the specific weight of the air–hydrometeor mixture. For the case where the domain is the half space z > 0 above flat ground, effective buoyancy β(x) at a given field point x is given by β(x) = −∞ −∞ −∞ [g2HρT()]/(4π|x|) dx̂ dẑ, where T is the total weight of air and hydrometeors per unit volume, the ≡ (x̂, ŷ, ) are source points, 2H is the horizontal Laplacian with respect to , and 2HρT(x̂, ŷ,) ≡ −2HρT(x̂, ŷ, ).

Corresponding author address: Dr. Robert Davies-Jones, NOAA/National Severe Storms Laboratory, 1313 Halley Circle, Norman, OK 73069-8493. Email: Bob.Davies-Jones@noaa.gov

1. Introduction

The question of whether certain air parcels near the ground can rise significantly is crucial to convective initiation and the dynamics of severe convective storms and tornadoes. For example, tornadogenesis may seem imminent based on radar observations but will not occur if surface-based parcels (below the radar horizon) are too negatively buoyant for a vortex aloft to draw them inward and upward (Leslie and Smith 1978; Markowski et al. 2002, 2003). Buoyancy normally is measured relative to conditions (or “the environment”) ahead of the storm and its cold pool. Note “cool air that is negatively buoyant relative to the environment may act as if it is ‘positively buoyant’ (i.e., rise) when in close proximity to very cold air” (Trapp and Davies-Jones 1997, p. 130).

Buoyancy in inhomogeneous surroundings requires definition. According to Glickman (2000), buoyancy is the upward force exerted upon a parcel of fluid in a gravitational field by virtue of the density difference between the parcel and that of the surrounding fluid. The Archimedean buoyancy force acting on an object is the weight of the fluid displaced minus the weight of the object. In atmospheric surroundings that vary horizontally, the weight of the displaced air is ill defined and so Archimedean buoyancy per unit volume depends on the local specific weight (weight per unit volume) and the specific weight at the same level in a subjectively chosen environment. In convection, there is a substantial vertical perturbation pressure-gradient force (VPPGF) in addition to buoyancy. Since the net vertical force is independent of the base state, the VPPGF is also relative to an arbitrary environment. Even when the environment is well-defined (horizontally uniform) and static, a substantial part of the buoyancy force is negated by an adverse VPPGF in the Archimedean formulation (Soong and Ogura 1973).

Moist convection is complicated by the presence of hydrometeors. For many purposes, a mixture of air and hydrometeors falling at their terminal velocities may be regarded as a single heterogeneous system with a system density (Emanuel 1994, p. 112). With this assumption, we can define a locally hydrostatic, upward pressure-gradient force per unit volume that balances the specific weight of the hydrometeors (the hydrometeor drag force) and the specific weight of moist air. In frictionless flow, the net vertical force is then the locally nonhydrostatic, vertical pressure-gradient force (NHVPGF). This formulation is non-Archimedean because there is no reference to a basic state (Das 1979). In the anelastic approximation, the NHVPGF can be decomposed into dynamically and statically forced parts (Davies-Jones 2002, hereafter DJ). To circumvent buoyancy's relativity and partial cancellation by the VPPGF, we can define the effective buoyancy force per unit volume to be statically forced part of the NHVPGF. Effective buoyancy is absolute because it depends only on the specific-weight field itself (see below) rather than on specific-weight perturbations from a basic state.

A similar decomposition of the VPPGF has been done for many years by numerical modelers in order to diagnose the dynamics of convective storms (as reviewed in Weisman and Rotunno 2000). When referred to a common entity (either unit mass or unit volume), the dynamically forced parts of the VPPGF and the NHVPGF are equal, while the effective buoyancy equals what Emanuel calls “the buoyant contribution,” which is defined as the sum of buoyancy and the statically forced part of the VPPGF (Emanuel 1994, p. 385). Modelers present plots of buoyant contribution without providing insight into its physical significance.

The force decompositions have facilitated physical understanding of phenomena. For example, lift provided by the dynamically forced part is largely responsible for the propagation of supercell updrafts (e.g., Weisman and Rotunno 2000; DJ) and is very important in the descent of tornadic vortices from aloft (Trapp and Davies-Jones 1997). Negative effective buoyancy near the ground influences updraft propagation at low levels when a strong cold pool is present (DJ) and inhibits tornado formation in very occluded mesocyclones (Markowski et al. 2002, 2003).

This paper finds an explicit relationship between effective buoyancy β (or buoyant contribution per unit volume) at a point and a weighted integral of the horizontal Laplacian of the specific weight of the air–hydrometeor mixture at surrounding points. This formula enables a relatively simple physical interpretation of β.

2. Formula for effective buoyancy

We assume that the flow is anelastic so that we can (a) eliminate rapid variations in pressure due to sound waves while retaining slower variations associated with convection, and (b) neglect sources and sinks of water vapor, due to phase changes, that are present in the continuity equation for moist air (Bannon 2002). In the non-Archimedean formulation (Das 1979; DJ), inviscid anelastic flow on an f plane is governed by the momentum and continuity equations
i1520-0469-60-23-2922-e1
where t is time; ∂t ≡ ∂/∂t, etc.; xxi + yj + zk ≡ (x1, x2, x3) is the position vector; k is the unit upward vector; ∇ ≡ ix + jy + kz; ∇Hix + jy; v ≡ (u, υ, w) ≡ (u1, u2, u3) is the velocity; ρ is the density of moist air; ρT is the mass of moist air and hydrometeors per unit volume (the system density); ρ(z) is the density of a reference state; pph + pnh is the pressure; pnh is the local nonhydrostatic pressure; phgMz T dz is the local hydrostatic pressure; and M is the mass of the column of unit cross section overlying the field point x. Note that −∂zphk balances the hydrometeor drag −g(ρTρ)k plus the specific weight of moist air. The vertical equation of motion reduces to ρdw/dt = −∂zpnh, so the NHVPGF vanishes at flat ground (z = 0). An entropy equation and continuity equations for the mixing ratios of the various hydrometeor species complete the system of equations.
Using (2), (1) may be written in its flux form:
i1520-0469-60-23-2922-e3
where the second term on the left-hand side, the divergence of the momentum-flux tensor ρuiuj, is expressed in tensor notation for simplicity. Taking the 3D divergence of (3) and again using (2) yields
i1520-0469-60-23-2922-e4
where ∂2(ρuiuj)/∂xixjρFDN(x) is the dynamical forcing of pnh.
We obtain an expression for effective buoyancy as follows. Taking −∂z of (4) yields
i1520-0469-60-23-2922-e5
Because (5) is linear in −∂zpnh, its solution can be decomposed into a part −∂zpnhDN that depends on the dynamical forcing function − ∂zFDN, and a part β that depends on purely static forcing arising from horizontal variations in the mass field (as in DJ). For flows that are in hydrostatic balance, the static and dynamic forcing terms in (5) cancel one another. From the static part of (5), β is the solution of
2βg2HρT
The absence of ρ(z) in (6) means that effective buoyancy is insensitive to the choice of reference-state density so long as ∇·(ρv) = 0 is a good approximation to mass conservation. Since the NHVPGF vanishes at flat ground, we assume that its parts, β and −∂zpnhDN, also vanish there. If the domain is the half space z > 0, the only boundary condition is the homogeneous Dirichlet one, β = 0 at z = 0. Thus, we can use the method of images (see p. 272 of Duff and Naylor 1966, hereafter DN) to extend the problem to the full space −∞ < x, y, z < ∞ by setting 2HρT(x, y,z, t) = −2HρT(x, y, z, t). The formal solution of (6) is
i1520-0469-60-23-2922-e7a
where r ≡ |x| is the distance to the field point x from a source or image point , 1/4πr is the Green's function for the full space and 2H ≡ ∂2/∂2 + ∂2/∂ŷ2 (DN, pp. 265, 272). An alternative version of (7a) is
i1520-0469-60-23-2922-e7b
where G(x, ) = 1/4πr − 1/4πr′ is the Green's function for the half space, and r′ ≡ [(x)2 + (yŷ)2 + (z + )2]0.5. The other piece of the vertical force, the part induced by motions, is given by
i1520-0469-60-23-2922-e8
At nonhorizontal boundaries the Neumann condition ∂β/∂n = −gn · ∇HρT applies to β [from n · ∂z of the static part of (1)]. Hence, effective buoyancy in a domain Ω with sloping terrain and/or side boundaries is given by
i1520-0469-60-23-2922-e9
where G(x, ) is now the Green's function for Ω, S is the bounding surface of Ω, and and are the area element and outward unit normal at for S (DN, p. 279).

3. Discussion

The standard buoyancy force per unit volume depends on a hypothetical environment and is equal to the deficit in system specific weight relative to the environment at the same height. The effective buoyancy force per unit volume βk is defined here as the statically forced part of the locally nonhydrostatic, upward pressure-gradient force. For inviscid anelastic flow in the half space z > 0 above flat ground, (7a) reveals that β is the potential associated with the source distribution g2HρT. In other words, β at a given field point is equal to the integral over the source and image points (with inverse-distance weighting) of the source strengths g2HρT(x). This conceptual relationship has not been discovered previously because its derivation in an Archimedean formulation is tortuous (see appendix). The quantity −ε2g2HρT() is proportional to the difference between the specific weight at and the average specific weight in a small neighborhood ( ± ε, ŷ ± ε, ) of or, alternatively, to the buoyancy at relative to the local environs. Thus, the effective buoyancy at a field point x is proportional to a weighted sum of the relative buoyancies at the source and image points . The influence of each remote relative buoyancy on effective buoyancy varies inversely with the distance |x|.

Acknowledgments

I am indebted to Dr. Peter Bannon for his helpful reviews, which resulted in a more rigorous analysis. This work was supported in part by NSF Grant ATM-0003869.

REFERENCES

  • Bannon, P., 2002: Theoretical foundations for models of moist convection. J. Atmos. Sci., 59 , 19671982.

  • Das, P., 1979: A non-Archimedean approach to the equations of convection dynamics. J. Atmos. Sci., 36 , 21832190.

  • Davies-Jones, R., 2002: Linear and nonlinear propagation of supercell storms. J. Atmos. Sci., 59 , 31783205.

  • Duff, G. F. D., , and D. Naylor, 1966: Differential Equations of Applied Mathematics. Wiley, 423 pp.

  • Emanuel, K. A., 1994: Atmospheric Convection. Oxford University Press, 580 pp.

  • Glickman, T. S., Ed.,. . 2000: Glossary of Meteorology. 2d ed. Amer. Meteor. Soc., 855 pp.

  • Leslie, L. M., , and R. K. Smith, 1978: The effect of vertical stability on tornadogenesis. J. Atmos. Sci., 35 , 12811288.

  • Markowski, P. M., , J. M. Straka, , and E. N. Rasmussen, 2002: Direct surface thermodynamic observations within the rear-flank downdrafts of nontornadic and tornadic supercells. Mon. Wea. Rev., 130 , 16921721.

    • Search Google Scholar
    • Export Citation
  • Markowski, P. M., , J. M. Straka, , and E. N. Rasmussen, 2003: Tornadogenesis resulting from the transport of circulation by a downdraft: Idealized simulations. J. Atmos. Sci., 60 , 795823.

    • Search Google Scholar
    • Export Citation
  • Soong, S-T., , and Y. Ogura, 1973: A comparison between axisymmetric and slab-symmetric cumulus cloud models. J. Atmos. Sci., 30 , 879893.

    • Search Google Scholar
    • Export Citation
  • Trapp, R. J., , and R. Davies-Jones, 1997: Tornadogenesis with and without a dynamic pipe effect. J. Atmos. Sci., 54 , 113133.

  • Weisman, M. L., , and R. Rotunno, 2000: The use of vertical wind shear versus helicity in interpreting supercell dynamics. J. Atmos. Sci., 57 , 14521472.

    • Search Google Scholar
    • Export Citation

APPENDIX

Derivation in the Archimedean formulation

Previous investigators used the Archimedean formulation and defined a buoyant contribution (per unit mass) to vertical acceleration (see Emanuel 1994, p. 385). The buoyant contribution per unit volume β̃ is defined as −gρT − ∂zpb, where ρT is the deviation of total density from the base-state density ρ(z); pb is the perturbation pressure owing to buoyancy; and −∂zpb is the solution of the boundary-value problem, −∇2(−∂zpb) = −gzzρT, −∂zpb = gρT at z = 0, in the case of a half-space domain. Although buoyant contribution and effective buoyancy are the same force (i.e., β̃ = β), the physical interpretation of this force has not been discovered previously because the direct explicit expression for β̃ is complicated. Adding the formal solution for −∂zpb (DN p. 265) to −gρT leads straight to the formula
i1520-0469-60-23-2922-ea1
where G(x, ) = 1/4πr − 1/4πr′ is the Green's function for the half space (DN, p. 272). Strangely, (A1) involves perturbation density even though β̃ itself does not depend on a base state. It is not obvious that (A1) simplifies to the explicit expression (7b) for effective buoyancy β(x). To see that it does, first note that
i1520-0469-60-23-2922-ea2
By Green's theorem,
i1520-0469-60-23-2922-ea3
But G(x, ) = 0 at = 0 and −2G(x, ) = δ(x) (DN, p. 265), where δ(x) is the Dirac function (DN, p. 40). Hence, the last three terms on the right-hand side of (A2) cancel and (A2) reduces to
i1520-0469-60-23-2922-ea4
which equals β(x) via (7b).
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