Article

Lamb's Hydrostatic Adjustment for Heating of Finite Duration

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Abstract

Lamb's hydrostatic adjustment problem for the linear response of an infinite, isothermal atmosphere to an instantaneous heating of infinite horizontal extent is generalized to include the effects of heating of finite duration. Three different time sequences of the heating are considered: a top hat, a sine, and a sine-squared heating. The transient solution indicates that heating of finite duration generates broader but weaker acoustic wave fronts. However, it is shown that the final equilibrium is the same regardless of the heating sequence provided the net heating is the same.A Lagrangian formulation provides a simple interpretation of the adjustment. The heating generates an entropy anomaly that is initially realized completely as a pressure excess with no density perturbation. In the final state the entropy anomaly is realized as a density deficit with no pressure perturbation. Energetically the heating generates both available potential energy and available elastic energy. The former remains in the heated layer while the latter is carried off by the acoustic waves.The wave energy generation is compared for the various heating sequences. In the instantaneous case, 28.6% of the total energy generation is carried off by waves. This fraction is the ratio of the ideal gas constant R to the specific heat at constant pressure cp. For the heatings of finite duration considered, the amount of wave energy decreases monotonically as the heating duration increases and as the heating thickness decreases. The wave energy generation approaches zero when (i) the duration of the heating is comparable to or larger than the acoustic cutoff period, 2/NA 300 s, and (ii) the thickness of the heated layer approaches zero. The maximum wave energy occurs for a thick layer of heating of small duration and is the same as that for the instantaneous case.The effect of a lower boundary is also considered.

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... How much derived an equation for the pressure as a function time does the adjustment require? of height in the final equilibrium state. For an In a series of papers Bannon (1995Bannon ( , 1996 isothermal atmosphere he found that the pressure below the heat source is unaffected by the heating, * e-mail: a.j.vandelden@phys.uu.nl whereas the pressure above the heat source is Tellus 52A (2000), 4 increased compared to the initial value immedi-2. The basic equations ately after the heating. ...
... The equa-nential decrease is proportional to the density scale height in an isothermal atmosphere, tions governing the vertical velocity (w), the potential temperature (h) and the Exner function (P ) H S ¬RT /g, where R is the ideal gas constant, T the temperature and g the acceleration due to are, assuming the atmosphere is an ideal gas ( Van Delden, 1992;Durran, 1999): gravity. This characteristic depth scale can, by analogy to the radius of deformation for geostrophic adjustment, be referred to as the radius Bannon (1996) extended these results to the nonlinear regime and to an atmosphere of arbitwhere time, t, and height, z are the independent rary thermal structure but, due to the method variables, J is the (diabatic) heating per unit mass, used, was unable to focus on the deformation scale per unit time and c v is the specific heat at constant height in these cases. Neither did he investigate in volume. ...
... The Exner function is related to the much detail the transient solution. Sotack and pressure, p, by: Bannon (1999) ...
... Previous investigations of geostrophic adjustment have partitioned the energy associated with a given imbalance among the buoyancy waves and steady state (e.g., Veronis 1956;Vadas and Fritts 2001). Likewise, investigations of hydrostatic adjustment have partitioned the energy among acoustic waves and the steady state (Bannon 1995;Sotack and Bannon 1999). The energy of the transients was inferred by taking the difference between the energy of the initial imbalance and that remaining in the steady state. ...
... Vadas and Fritts (2001) presented a similar conclusion with respect to Boussinesq buoyancy waves generated by time-dependent sources of momentum and heat. Similarly, Sotack and Bannon (1999) conclude that acoustic waves are not generated significantly by heating of duration exceeding 2 min. Addi-tionally, the comparison of the wave filtering function given by a top-hat injection to that of a sine-squared injection implies that an injection that is smoother in time will project less energy onto higher-frequency harmonics. ...
Article
This second part of a two-part study of the hydrostatic and geostrophic adjustment examines the potential vorticity and energetics of the acoustic waves, buoyancy waves, Lamb waves, and steady state that are generated following the prescribed injection of heat into an isothermal atmosphere at rest. The potential vorticity is only nonzero for the steady class and depends only on the spatial and time-integrated properties of the injection. The waves contain zero net potential vorticity, but undergo a time-dependent vorticity exchange involving latent and relative vorticities.The energy associated with a given injection may be partitioned distinctly among the various wave classes. The characteristics of this partitioning depend on the spatiotemporal detail of the injection, as well as whether the imbalance is generated by injection of heat, mass, or momentum. Spatially, waves of a scale similar to that of the injection are preferentially excited. Temporally, an extended duration injection preferentially filters high-frequency waves. An instantaneous injection, that is, the temporal Green's function, contains the largest proportions of the high-frequency waves.The proportions of kinetic, available elastic, and available potential energies that are carried by the various waves are functions of the homogeneous system. For example, deep buoyancy waves of small horizontal scale primarily contain equal portions of available potential and vertical kinetic energy. The steady state contains more available potential energy than kinetic energy at small horizontal scale, and vice versa. These qualities of the wave energetics illustrate the mechanisms that characterize the physics of each wave class.The evolution and spectral partitioning of the energetics following localized warmings identical to those in Part I are presented in order to illustrate some of these basic properties of the energetics. For example, a heating lasting longer than a few minutes does not excite acoustic waves. However, Lamb waves of wide horizontal scale can be excited by a heating of several hours. The first buoyancy waves to be filtered by an extended duration heating are those of the deepest and narrowest structure that have a frequency approaching the buoyancy frequency. The energetics of the steady state depends only on the spatial and time-integrated properties of the warming. However, the energetics and transient evolution toward a given steady state depend on the temporal properties of the warming and may differ widely.
... However, the effects of diabatic forcing in the nonhydrostatic global model can be investigated using the present normal modes. Sotack and Bannon (1999) discussed a prototype problem of hydrostatic adjustment, that is, how the hydrostatic equilibrium is established from a hydrostatically unbalanced state, resulting from time-dependent heating. Understanding of hydrostatic adjustment is a prerequisite to studying the problem of data initialization with the nonhydrostatic model just as in the case of geostrophic adjustment (e.g., Blumen 1972) in a primitive equation model. ...
Article
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Anticipating use of a very high resolution global atmospheric model for numerical weather prediction in the future without a traditional hydrostatic assumption, this article describes a unified method to obtain the normal modes of a nonhydrostatic, compressible, and baroclinic global atmospheric model. A system of linearized equations is set up with respect to an atmosphere at rest. An eigenvalue-elgenfunction problem is formulated, consisting of horizontal and vertical structure equations with suitable boundary eonditions. The wave frequency and the separation parameter, referred to as "equivalent height," appear in both the horizontal and vertical equations. Hence, these two equations must be solved as a coupled problem. Numerical results are presented for an isothermal atmosphere. Since the solutions of the horizontal structure equation can only be obtained numerically, the coupled problem is solved by an iteration method. In the primitiveequation (hydrostatic) models, there are two kinds of normal modes: The first kind consists of eastward and westward propagating gravity-inertia oscillations, and the second kind consists of westward propagating rotational (Rossby-Haurwitz type) oscillations. In the nonhydrostatic model, there is an additional kind of eastward and westward propagating acoustic-inertia oscillations. The horizontal structures of the third kind are distinguished from those of the first kind by large differences in the equivalent height. The second kind is hardly affected by nonhydrostatic effects. In addition, there are so-called external inertia-gravity mode oscillations (Lamb waves), which propagate horizontally with almost constant speed of sound. Also, there are external rotational mode oscillations that correspond to equivalent barotropic planetary waves. Those two classes of oscillations are identical to those in the hydrostatic version of model. Nonhydrostatic effects on the first kind of oscillations become significant for smaller horizontal and deeper vertical scales of motion.
... A longer duration forcing generates less total energy than a forcing applied rapidly. Similar results have been demonstrated analytically (see Sotack and Bannon 1999;Chagnon and Bannon 2005a). The loss of energy at longer duration is due to the destructive interference of waves whose period is shorter than the time scale of the forcing. ...
Article
The interaction between gravity waves and the mean flow through which they propagate has been studied extensively and continues to be an active area of research, particularly with regard to the dynamics of the middle atmosphere. A problem that is complimentary to the wave - mean flow interaction is that concerning the interaction between gravity waves and the forcing from which they originate. The wave - forcing interaction problem is of particular relevance to mesoscale dynamics of the troposphere where, for example, convective systems evolve on time and space scales similar to the gravity waves they spawn. This paper introduces the concept of dynamical resistance in order to facilitate the analysis of gravity wave - forcing interaction. The dynamical resistance is defined as the work performed by gravity waves upon a forcing. A few simple examples are provided in order to demonstrate the dependence of the dynamical resistance and forcing efficiency on the properties of the forcing as well as the environment against which the forcing is applied. It is shown that optimal configurations of the basic state and forcing exist which minimize the resistance imposed by gravity waves upon their source and thus maximize the total energy generated by the forcing. Calculations are performed both in a simple linear model and along hypothetical ray paths for an isolated forcing propagating relative to the background flow and for fields of multiple forcing elements.
... Examining shorter and longer duration heatings show that the amount of energy projected onto the waves is inversely proportional to the duration of the heating. These findings are in full agreement with Sotack and Bannon (1999) and Chagnon and Bannon (2005b). A larger spatial heating generates a greater amount of both available potential and available elastic energy. ...
Article
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... Lamb's (1932) problem considered the vertical adjustment by a compressible atmosphere to horizontally invariant perturbations. The numerous recent extensions of Lamb's problem (Bannon 1995;Sotack and Bannon 1999;Kalashnik 2000;Duffy 2003) offer a theory to explain how an atmosphere establishes and maintains hydrostatic balance. Essentially, a warmed compressible atmosphere must expand vertically in order to establish hydrostatic balance. ...
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