η = ζ + f
(4.85)
and note that the Coriolis parameter f is a function of latitude only, the rate of change of the absolute vorticity at a given point of the atmosphere is given by
(4.86)
The first two terms on the right-hand side of the equation represent the horizontal and vertical advection of the absolute vorticity, respectively. The third term, called the vortex stretching term, describes the stretching or compression of vortex tubes in the vertical direction that results from a non-zero horizontal convergence or divergence of the flow. The fourth term, known as the tilting term, describes how the horizontal variations in the vertical velocity tend to tilt the horizontal components of the absolute vorticity vector toward the vertical direction. The fifth term, called the solenoid term, describes the effect of barocliniticity on the vertical component of the vorticity, and the last term represents the effects of friction.
Simpler forms of the vorticity equation ignore the smallest terms in the equation on the basis of a scale analysis or as a result of specific assumptions. For example, if one assumes that the fluid is incompressible (divergence of the velocity is equal to zero) and barotropic (no horizontal variation of the temperature, no vertical variation of wind velocities, so that surfaces of equal pressure coincide with surfaces of equal density; see Section 2.8), and if one neglects friction forces, one obtains the following equation
(4.87)
which states that the rate of change of absolute vorticity following an air parcel is proportional to the convergence of the horizontal velocity. The first computer weather forecast by Charney et al. (1950) was based on the barotropic vorticity equation. Such models suffer from some severe restrictions (such as their inability to correctly generate cyclones), so that later models have been expressed by conserving the baroclinic terms in the governing equations.
Large-scale flows in the extratropics can also be represented by the quasi-geostrophic vorticity equation. This equation is established by decomposing the horizontal wind velocity vh into a geostrophic component vg (see Section 2.7 for its definition) and a remaining ageostrophic component vag of smaller amplitude. The actual wind in the momentum equation is approximated by its geostrophic component, except in the case of the divergence term. This exception (which explains why the approximation is known as the quasi-geostrophic approximation) is justified by the fact that the divergence of the geostrophic wind velocity is equal to zero and that, without accounting for the non-zero divergence of the ageostrophic wind component, no vertical wind could be generated. The resulting equation for the quasi-geostrophic relative vorticity, in which friction is ignored, is
(4.88)
or, if the fluid is assumed to be incompressible,
(4.89)
These two equations are based on the mid-latitude beta-plane approximation in which the Coriolis parameter f = f0 + βy is assumed to vary linearly with the geometric distance y measured along a meridian, and f0 = 2 Ω sin φ0 denotes the value of parameter f at a point of reference at latitude φ0 where y = 0.
Following Helmholtz’s theorem, the horizontal wind vector vh can be deduced from the vorticity and divergence by noting that it can be separated into two components
vh = [k × ∇hΨ] + ∇hχ
(4.90)
where Ψ is the streamfunction (which represents the nondivergent part of the flow whose value is constant along a streamline that follows the flow) and χ is the velocity potential (a scalar function whose gradient equals the velocity of an irrotational flow). It is straightforward to show that these scalar terms are related to the vorticity ζ and divergence D by
(4.91)
(4.92)
where the “horizontal” Laplacian operator is defined in spherical coordinates as
(4.93)
By combining relations (4.91) and (4.92) with the vorticity and divergence equations (4.81) and (4.82), one derives easily the two components of the horizontal wind velocity from the calculated streamfunction and velocity potential. The vertical wind component is obtained from the continuity equation. The potential vorticity, a conserved quantity in the absence of dissipative processes, is often used to diagnose atmospheric transport (see Box 4.3).
Box 4.3 Relative, Absolute, and Potential Vorticity
The relative vorticity ζ defined in (4.83) is a measure of the spin of a fluid parcel relative to coordinates attached to the Earth. The absolute vorticity ζ + f is the sum of the spin of the fluid and the planetary vorticity (represented here by the Coriolis parameter f). It is not a conserved quantity since, even in the absence of dissipation, its tendency is proportional to the divergence of the motion field.
In the case of an incompressible fluid and in the absence of dissipative forces, the integration of the continuity equation over a height h separating two free surfaces of a fluid (often referred to as the depth of the fluid), shows that the potential vorticity
is conserved following the motion of a fluid parcel. This property implies that the relative vorticity must adjust in response to changes in the planetary vorticity (as the parcel is displaced in latitude) and to changes in the depth of the fluid.
For a compressible fluid such as air, it is possible to derive a similar conservation principle for the Ertel potential vorticity [m2 s–2 K kg–1]
where ρa is the air density, Ω is the angular velocity vector of the Earth’s rotation, v the three-dimensional velocity field, and θ the potential temperature. Neglecting some minor terms, P is expressed in spherical coordinates as
where a is the Earth’s radius. In the absence of friction and heat sources or sinks, the Ertel potential vorticity P is a materially conservative property. As a result, P is often used to diagnose transport processes in the atmosphere since, over relatively short timescales during which diabatic and other dissipative processes can be neglected, this quantity is an excellent tracer of fluid motion. It has become accepted to define 1.0 × 10−6 m2 s−1 K kg−1 as one potential vorticity unit (1 PVU).
In the upper atmosphere, above approximately 100 km altitude, molecular viscosity (as measured by the viscosity coefficient μ) must be taken into account. Collisions between neutral particles and ions, whose motions are sensitive to electromagnetic fields, also affect the winds, especially at high latitudes. The effect of this ion drag is often assumed to be proportional to the difference between the neutral and ion winds. Thus, if vion represents the ion velocity (bulk motions and gyromotions generated by the electromagnetic fields) and νion the ion-neutral collision frequency [s–1], the momentum equation (4.71) becomes
(4.94)
where, as above, v is the neutral wind velocity, Ω the angular velocity of the Earth rotation vector, p the total pressure, and z the altitude.
4.5.3 Energy
The equation of energy is an expression of the first law of thermodynamics. It states that the energy supplied to an air parcel produces an increase in its internal energy (cvT) or induces work by expansion. Thus per unit time, the energy conservation is expressed as
(4.95)
where Q [J kg–1 s–1] is the diabatic heating term and cv [J K–1 kg–1] the specific heat at constant volume. Diabatic heating/cooling may be driven by the absorption/emission of radiative energy by atmospheric gases, or by condensation/evaporation of water vapor. The value of cv for air at 0 °C is 717.5 J kg–1 K–1. The second term in this expression accounts for the work done upon a unit mass of air by compression or expansion of the volume. The reciprocal density 1/ρa is termed the specific volume [m3 kg–1]. When using the ideal gas approximation, this energy conservation can be expressed as
(4.96)
where cp [J K–1 kg–1] is the specific heat at constant pressure (cp = cv + R where R is the gas constant). Its value at 0 °C is thus 717.5 + 287 = 1004.5 J K–1 kg–1. The second term expresses compression heating or expansion cooling associated with adiabatic processes taking place in the compressible fluid.
Making use of the hydrostatic equation (4.75), the energy equat
ion becomes
(4.97)
where H is the scale height. In spherical coordinates,
(4.98)
In the absence of significant diabatic processes (Q = 0) and horizontal temperature gradients, the vertical temperature profile is obtained from
(4.99)
It is often useful to introduce the potential temperature θ, defined in Chapter 2,
(4.100)
whose vertical gradient ∂θ/∂z = 0 under adiabatic conditions. θ is a better marker of diabatic processes and atmospheric heat transport than the absolute temperature T. In this case, the energy equation takes the form
(4.101)
where the ratio R/cp = 0.285. This equation shows that, in the absence of diabatic processes, the potential temperature of an air parcel is a conserved quantity along the motion of the fluid. In spherical coordinates (4.101) becomes
(4.102)
When heating and cooling are produced by radiative absorption and emission, the value of the net heating rate Q is derived from the divergence of the radiative flux calculated at each point of the atmosphere after spectral integration. See Chapter 5 for more details.
In the upper atmosphere, additional processes affect the energy budget and must be accounted for in the energy conservation equation. In the thermosphere, heat transport by conduction becomes important. Diabatic heating that must be considered explicitly includes absorption of shortwave solar radiation, chemical heating resulting from collisional deactivation of energetically excited species, and chemical heating from exothermic reactions. Joule heating, which arises from the dissipation of electric currents in the ionosphere, represents another important effect, especially during geomagnetic storms. Infrared emissions of atomic oxygen and nitric oxide are important cooling processes.
4.5.4 Primitive and Non-Hydrostatic Equations
Meteorological models are based on the equations presented in the previous sections. If, in these equations, one separates the horizontal and vertical components of the velocity vector (v = vh + wk), and assumes that the atmosphere is frictionless, the non-hydrostatic equations needed to derive the dependent variables vh (u,v), w, p, ρ and T are
(4.103)
(4.104)
(4.105)
(4.106)
p = ρaRT equation of state
(4.107)
This system of equations describes air motions in a dry atmosphere over a wide range of scales, including the propagation of planetary waves, inertia-gravity waves, and even acoustic (sound) waves. An additional equation expressing the mass conservation of water vapor is added to fully describe a moist atmosphere:
where μw is the mass mixing ratio of water vapor (usually referred to as specific humidity and denoted q in the meteorological literature). The source and sink terms E and C account for the evaporation and condensation processes and represent the influence of physical processes on the dynamics of the atmosphere.
With the exception of the equation of state, which is a diagnostic relation that relates the pressure, density, and temperature, all other equations are prognostic equations that can be integrated forward in time and predict the evolution of atmospheric variables if the initial meteorological situation is known.
When the vertical acceleration dw/dt can be neglected in comparison with other terms, the differential equation for the vertical velocity (4.105) is replaced by the diagnostic hydrostatic balance approximation (4.75). The resulting equations are referred to as the primitive equations because they are very close to the original system established in 1904 by Vilhelm Bjerknes, and on which numerical weather predictions were first attempted.
One of the difficulties encountered in the early integration of the dynamical equations was the generation of large-amplitude, fast-propagating waves (acoustic and gravity waves) of no meteorological significance that resulted from a lack of momentum balance in the adopted initial conditions. This issue explains why Lewis Fry Richardson failed in his attempt to provide a successful numerical weather forecast in the late 1910s: high-frequency acoustic waves generated large time derivatives in the numerical method that masked the time derivatives associated with the actual weather signal (see Section 1.4). Thirty years later, in 1950, Jule Charney, Agnar Fjörtoft, and John von Neumann solved the hydrostatic equations of motions by imposing an additional quasi-geostrophic approximation. This configuration leads to the elimination of acoustic and gravity waves, but retains larger patterns such as the Rossby (planetary) waves and mesoscale weather systems. Quasi-geostrophic models, known also as filtered-equation models, do not require that initial conditions be perfectly balanced.
The geostrophic approximation adopted in the early models is now regarded as inaccurate and unnecessarily restrictive; in addition, it is invalid in the tropics and therefore must be reserved for examining simple scientific questions in the extra-tropics. Many global models of the atmosphere are based on the primitive equations using the hydrostatic approximation, which is convenient for treating motions at horizontal scales larger than about 10 km. A disadvantage of the hydrostatic models is that they cannot be applied to simulate small-scale processes such as convection, storms, or mountain waves, which are characterized by large vertical accelerations. High-resolution weather prediction and even global atmospheric models are increasingly based on non-hydrostatic equations in which the full vertical momentum equation (4.105) is retained.
Non-hydrostatic models are considerably more complex and computationally more demanding than the hydrostatic models. In addition, they generate fast-propagating acoustic waves, whose propagation depends on the compressibility of the fluid. These waves must be eliminated in order to avoid the use of prohibitively small time steps to solve the equations. This can be achieved by adopting an incompressibility assumption (dρa/dt = 0), which is appropriate for shallow atmospheric circulations (vertical depth of the motion considerably smaller than the typical horizontal scale of the circulation). In this case, the continuity equation reduces to the prognostic equation
∇h ⋅ vh + ∂w/∂z = 0
(non-divergent wind). Equations for shallow layers are often considered as prototypes for the primitive equations, and used to test numerical methods.
Another approach to filter acoustic waves, not limited to shallow flows, is to assume that the air density varies with height (denoted ρ0(z)), but does not change locally with time (∂ρa/∂t = 0). In this case, the continuity equation (4.103) for fully compressible (elastic) air is replaced by the diagnostic anelastic continuity equation
∇h ⋅ (ρ0vh + ∂(ρ0w)/∂z) = 0
.
This particular approximation has been widely adopted for non-hydrostatic atmospheric models.
Finally, in the Boussinesq approximation, which also eliminates acoustic waves, the variables representing density and pressure in the equations of motions are separated into a reference state in hydrostatic balance, and small perturbations from this basic state. In the resulting equations, these perturbations are neglected, except in the buoyancy (gravity) term of the momentum equation.
4.6 Vertical Coordinates
So far we have expressed the equations for chemical continuity and atmospheric dynamics using the geometric altitude z [m] as vertical coordinate. This is the obvious choice for vertical coordinate but generally not the best. Other vertical coordinates can be used that are single-valued, monotonic functions of z. This is the case, for example, of the atmospheric pressure p (under the hydrostatic approximation). The following rule (Kasahara, 1974) allows us to transform the equations to arbitrary coordinates η(λ, φ, z, t) defined as a function of longitude λ, latitude φ, altitude z, and time t. For any dependent variable Ψ(λ, φ, z, t), we write (Lauritzen et al., 2011):
(4.108)
where s can be λ, φ, or t. We deduce that
(4.109)
Similarly
(4.110)
The total derivative of Ψ is expressed as
(4.111)
where vη is the projection
of the velocity on a η-surface, is the “vertical” velocity in the η coordinate system, and ∇η is the “horizontal” gradient calculated on a surface of constant η. The total derivative dΨ/dt is a property of the fluid and is therefore independent of the coordinate system.
Different vertical coordinates that can be considered besides the geometric altitude include (1) pressure coordinates, particularly attractive in the case of hydrostatic models, (2) terrain-following coordinates in which the application of lower boundary conditions is facilitated, (3) hybrid terrain-following variants that are terrain-following in the lowest levels of the atmosphere and pressure-following earlier, and (4) isentropic coordinates that approximate Lagrangian coordinates when diabatic processes are weak. We review these different coordinate systems next.
4.6.1 Pressure Coordinate System
Meteorologists often use atmospheric pressure to express the vertical variations of the dependent variables. This is the pressure or isobaric coordinate system. It is the most universally used coordinate in dynamical models since, under hydrostatic assumptions, the forms of some of the dynamical equations become particularly simple. Already at the beginning of the twentieth century, Vilhelm Bjerknes was establishing his synoptic charts on isobaric surfaces, and today meteorological variables (temperature, geopotential, etc.) are often represented as a function of atmospheric pressure. The relation between pressure and altitude is expressed by the hydrostatic approximation
dp = – ρa g dz
Modeling of Atmospheric Chemistry Page 17