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Abstract
Modified Fourier series, as judged by criteria of accuracy, numerical efficiency and ease of programming, are the best choice of latitudinal expansion functions for general problems on the sphere. The pseudospectral and spectral methods, however, can be easily and successfully applied with all three types of orthogonal series. For special situations, such as when the latitude variable is stretched, Chebyshev polynomials are the only practical choice, but for orthodox problems on the globe, they are less efficient than the other two sets of functions. Although spherical harmonics have been universally employed in the past, Fourier series give comparable accuracy and are significantly easier to program and manipulate. Thus, in the absence of a special reason to the contrary, the simplest and most effective way to handle the north–south dependence of the solution to a boundary or eigenvalue problem on the sphere is to use a Fourier series in colatitude.
Abstract
Modified Fourier series, as judged by criteria of accuracy, numerical efficiency and ease of programming, are the best choice of latitudinal expansion functions for general problems on the sphere. The pseudospectral and spectral methods, however, can be easily and successfully applied with all three types of orthogonal series. For special situations, such as when the latitude variable is stretched, Chebyshev polynomials are the only practical choice, but for orthodox problems on the globe, they are less efficient than the other two sets of functions. Although spherical harmonics have been universally employed in the past, Fourier series give comparable accuracy and are significantly easier to program and manipulate. Thus, in the absence of a special reason to the contrary, the simplest and most effective way to handle the north–south dependence of the solution to a boundary or eigenvalue problem on the sphere is to use a Fourier series in colatitude.
Abstract
It is shown that spectral and pseudospectral methods can lead to great savings in numerical efficiency-typically, at least a factor of four in storage and eight in operation count-in comparison with finite-difference methods for solving eigenvalue and nonseparable, noniterative boundary value problems at the cost of only a a slight amount of additional programming. The special problems or apparent problems that geophysical boundary layers, critical latitudes and critical levels and real data create for spectral algorithms are also discussed along with when and how these can be solved.
Abstract
It is shown that spectral and pseudospectral methods can lead to great savings in numerical efficiency-typically, at least a factor of four in storage and eight in operation count-in comparison with finite-difference methods for solving eigenvalue and nonseparable, noniterative boundary value problems at the cost of only a a slight amount of additional programming. The special problems or apparent problems that geophysical boundary layers, critical latitudes and critical levels and real data create for spectral algorithms are also discussed along with when and how these can be solved.
Abstract
Regional spectral models have previously periodized and blended limited-area data through ad hoc low-order schemes justified by intuition and empiricism. By using infinitely differentiable “window functions” or “bells” borrowed from wavelet theory, one can periodize with preservation of spectral accuracy. Similarly, it is shown through a mixture of theory and numerical examples that Davies relaxation for blending limited-area and global data in one-way nested forecasting can be performed using the same C ∞ bells as employed for the Fourier blending.
“The relative success of empirical methods . . . may be used as partial justification to allow us to make the daring approximation that the data on a limited area domain may be decomposed into a trend and a periodic perturbation, and to proceed with Fourier transformation of the latter.” Laprise (2003, p. 775)
Abstract
Regional spectral models have previously periodized and blended limited-area data through ad hoc low-order schemes justified by intuition and empiricism. By using infinitely differentiable “window functions” or “bells” borrowed from wavelet theory, one can periodize with preservation of spectral accuracy. Similarly, it is shown through a mixture of theory and numerical examples that Davies relaxation for blending limited-area and global data in one-way nested forecasting can be performed using the same C ∞ bells as employed for the Fourier blending.
“The relative success of empirical methods . . . may be used as partial justification to allow us to make the daring approximation that the data on a limited area domain may be decomposed into a trend and a periodic perturbation, and to proceed with Fourier transformation of the latter.” Laprise (2003, p. 775)
Abstract
Using a simple separable model in which the mean wind U(y) is assumed to be a function of latitude only, those effects of latitudinal shear which do not depend on the vanishing of U(y) are examined for planetary waves in the middle atmosphere (the stratosphere and mesosphere).
First, it is shown that for nonsingular wind profiles the WKB method and ray tracing may be inaccurate for meridional shear. It is both physically and mathematically preferable to interpret the results of more complex models in terms of vertically propagating modes since the amplitude of the waves as a function of latitude is determined primarily by the modal structure rather than by variations of the mean wind or the refractive index.
Second, it is demonstrated that westerly planetary gravity waves, which are vertically trapped as shown by Charney and Drazin (1961), are also latitudinally trapped near the pole where the mean winds are easterly. In consequence, such waves, which form the quasi-stationary spectrum of the summer hemisphere, are unaffected by the critical latitude in the subtropics of the winter hemisphere. The physical implications of these and other findings are discussed.
Abstract
Using a simple separable model in which the mean wind U(y) is assumed to be a function of latitude only, those effects of latitudinal shear which do not depend on the vanishing of U(y) are examined for planetary waves in the middle atmosphere (the stratosphere and mesosphere).
First, it is shown that for nonsingular wind profiles the WKB method and ray tracing may be inaccurate for meridional shear. It is both physically and mathematically preferable to interpret the results of more complex models in terms of vertically propagating modes since the amplitude of the waves as a function of latitude is determined primarily by the modal structure rather than by variations of the mean wind or the refractive index.
Second, it is demonstrated that westerly planetary gravity waves, which are vertically trapped as shown by Charney and Drazin (1961), are also latitudinally trapped near the pole where the mean winds are easterly. In consequence, such waves, which form the quasi-stationary spectrum of the summer hemisphere, are unaffected by the critical latitude in the subtropics of the winter hemisphere. The physical implications of these and other findings are discussed.
Abstract
With the simplifying assumption that the mean zonal wind is a function of latitude only, numerical and analytical methods are applied to study the effects of critical latitudes (where the Doppler-shifted frequency is 0) on planetary waves. On the midlatitude beta-plane, it is shown that the modes divide into two limiting classes. The low-order, vertically propagating modes are confined to that side of the critical latitude where the mean winds are westerly as found by Dickinson (1968b). The high-order modes, although vertically trapped, are indifferent to the singularity and oscillate sinusoidally on both sides of the critical latitude as if it were not present. On the sphere, there is also a third class of low-order modes which are latitudinally trapped near the pole where the winds are easterly and also are unaffected by the critical latitude. Numerical studies show that it is the location of the critical latitude far more than the intensity or shape of the winds that controls the dynamics of the low-order, vertically propagating modes.
The most striking conclusion on the equatorial beta-plane is that sufficiently strong linear shear, although stable by conventional criterion, makes the Kelvin wave unstable. Together with the transparency of the high-order global modes, this shows that strong baroclinity may drastically alter the behavior of waves with critical latitudes, from that predicted by the barotropic or near-barotropic models so widely applied to critical latitudes in the past.
Abstract
With the simplifying assumption that the mean zonal wind is a function of latitude only, numerical and analytical methods are applied to study the effects of critical latitudes (where the Doppler-shifted frequency is 0) on planetary waves. On the midlatitude beta-plane, it is shown that the modes divide into two limiting classes. The low-order, vertically propagating modes are confined to that side of the critical latitude where the mean winds are westerly as found by Dickinson (1968b). The high-order modes, although vertically trapped, are indifferent to the singularity and oscillate sinusoidally on both sides of the critical latitude as if it were not present. On the sphere, there is also a third class of low-order modes which are latitudinally trapped near the pole where the winds are easterly and also are unaffected by the critical latitude. Numerical studies show that it is the location of the critical latitude far more than the intensity or shape of the winds that controls the dynamics of the low-order, vertically propagating modes.
The most striking conclusion on the equatorial beta-plane is that sufficiently strong linear shear, although stable by conventional criterion, makes the Kelvin wave unstable. Together with the transparency of the high-order global modes, this shows that strong baroclinity may drastically alter the behavior of waves with critical latitudes, from that predicted by the barotropic or near-barotropic models so widely applied to critical latitudes in the past.
Abstract
Andrews and Hoskins used semigeostrophic theory to argue that the energy spectrum of a front should decay like the −8/3 power of the wavenumber. They note, however, that their inviscid analysis is restricted to the very moment of breaking; that is, to the instant t = t β when the vorticity first becomes infinite. In this paper, Burgers' equation is used to investigate the postbreaking behavior of fronts. We find that for t > t β , the front rapidly evolves to a jump discontinuity. Combining our analysis with the Eady wave/Burgers„ study of Blumen, we find that the energy spectrum is more accurately approximated by the −8/3 power of the wavenumber, rather than by the k −2 energy spectrum of a discontinuity, for less than two hours after the time of breaking.
We also offer two corrections. Cai et al. improve a pseudospectral algorithm by fitting the spectrum of a jump discontinuity. This is not legitimate at t = t β because the front initially forms with a cube root singularity and its spectral coefficients decay at a different rate. Whitham claims that for t > t β, the characteristic equation has two roots. We show by explicit solution that there are actually three.
Abstract
Andrews and Hoskins used semigeostrophic theory to argue that the energy spectrum of a front should decay like the −8/3 power of the wavenumber. They note, however, that their inviscid analysis is restricted to the very moment of breaking; that is, to the instant t = t β when the vorticity first becomes infinite. In this paper, Burgers' equation is used to investigate the postbreaking behavior of fronts. We find that for t > t β , the front rapidly evolves to a jump discontinuity. Combining our analysis with the Eady wave/Burgers„ study of Blumen, we find that the energy spectrum is more accurately approximated by the −8/3 power of the wavenumber, rather than by the k −2 energy spectrum of a discontinuity, for less than two hours after the time of breaking.
We also offer two corrections. Cai et al. improve a pseudospectral algorithm by fitting the spectrum of a jump discontinuity. This is not legitimate at t = t β because the front initially forms with a cube root singularity and its spectral coefficients decay at a different rate. Whitham claims that for t > t β, the characteristic equation has two roots. We show by explicit solution that there are actually three.
Abstract
For linearized hydrostatic waves on a spherical earth with a zonal mean wind which is a function of latitude and pressure I derive, without further approximations, expressions for the vertical and meridional energy fluxes in terms of the meridional heat flux and the vertical and meridional fluxes of zonal momentum. Using these expressions, I prove that in the absence of critical surfaces, dissipation, thermal heating and nonharmonic time dependence, that the waves and mean flow do not interact: the wave Reynold's stresses are exactly balanced by a mean meridional circulation whose streamfunction is simply the meridional beat flux divided by the static stability. In the presence of dissipation, thermal heating or transience, 1 am able to express the net forcing of the mean blow by the waves as expressions which are explicitly proportional to the coefficients of dissipation and heating and to the imaginary part of the phase speed. My work significantly extends earlier theorems on the noninteraction of waves with the zonally averaged flow and on the interrelationships of wave fluxes proved by Eliassen and Palm, Charney and Drazin, and Holton because my theorems eliminate some important restrictive assumptions and include all these previous results as special cases.
Abstract
For linearized hydrostatic waves on a spherical earth with a zonal mean wind which is a function of latitude and pressure I derive, without further approximations, expressions for the vertical and meridional energy fluxes in terms of the meridional heat flux and the vertical and meridional fluxes of zonal momentum. Using these expressions, I prove that in the absence of critical surfaces, dissipation, thermal heating and nonharmonic time dependence, that the waves and mean flow do not interact: the wave Reynold's stresses are exactly balanced by a mean meridional circulation whose streamfunction is simply the meridional beat flux divided by the static stability. In the presence of dissipation, thermal heating or transience, 1 am able to express the net forcing of the mean blow by the waves as expressions which are explicitly proportional to the coefficients of dissipation and heating and to the imaginary part of the phase speed. My work significantly extends earlier theorems on the noninteraction of waves with the zonally averaged flow and on the interrelationships of wave fluxes proved by Eliassen and Palm, Charney and Drazin, and Holton because my theorems eliminate some important restrictive assumptions and include all these previous results as special cases.
Abstract
The slow manifold of an inviscid five-mode model introduced by Lorenz is investigated. When the influence of the gravity modes on the Rossby modes is neglected, the analytical solution given by Lorenz and Krishnamurthy is generalized. When gravity-Rossby coupling is included, direct numerical solutions are computed by solving a nonlinear boundary value problem. In all cases, the slow manifold has gravity mode oscillations that mimic free gravity waves and whose amplitude is proportional to the exponential of the reciprocal of the Rossby number ε.
Abstract
The slow manifold of an inviscid five-mode model introduced by Lorenz is investigated. When the influence of the gravity modes on the Rossby modes is neglected, the analytical solution given by Lorenz and Krishnamurthy is generalized. When gravity-Rossby coupling is included, direct numerical solutions are computed by solving a nonlinear boundary value problem. In all cases, the slow manifold has gravity mode oscillations that mimic free gravity waves and whose amplitude is proportional to the exponential of the reciprocal of the Rossby number ε.
Abstract
By using the method of multiple scales in height and a variety of methods in latitude, analytic solutions for equatorial waves in combined vertical and horizontal shear are derived. In contrast to the formulation of Andrews and McIntyre (1976b), latitudinal shear is incorporated at lowest order in the vertical shear expansion, showing that it is unnecessary to carry the calculation to first explicit order in Ri−½, where Ri is the Richardson number. The multiple-scales approximation implies that, with the exception of the overall amplitude factor and arbitrary overall constant phase factor, all properties of the wave at the height z=z 0 are determined by the local wind profile, V(y)=U(y,z 0). In consequence, understanding waves in two-dimensional shear reduces to the much simpler problem of solving the one-dimensional eigenvalue equation in latitude which is derived by assuming that the mean wind is V(y), a function of latitude only. This is done using ordinary perturbation theory, a non-perturbative analytic procedure and the Hermite spectral method for various classes of waves. Once its solutions are known, the overall amplitude factor may be found by using the wave action equation as shown in the text. When the method of multiple scales is invalid, as appears true of the tropical ocean, it is shown that Hermite spectral methods in latitude are much more accurate (at least in the absence of coastal boundaries) than the finite-difference methods used in the past. The techniques discussed here are applied to several classes of observed atmospheric equatorial waves in Part II (Boyd, 1978a).
Abstract
By using the method of multiple scales in height and a variety of methods in latitude, analytic solutions for equatorial waves in combined vertical and horizontal shear are derived. In contrast to the formulation of Andrews and McIntyre (1976b), latitudinal shear is incorporated at lowest order in the vertical shear expansion, showing that it is unnecessary to carry the calculation to first explicit order in Ri−½, where Ri is the Richardson number. The multiple-scales approximation implies that, with the exception of the overall amplitude factor and arbitrary overall constant phase factor, all properties of the wave at the height z=z 0 are determined by the local wind profile, V(y)=U(y,z 0). In consequence, understanding waves in two-dimensional shear reduces to the much simpler problem of solving the one-dimensional eigenvalue equation in latitude which is derived by assuming that the mean wind is V(y), a function of latitude only. This is done using ordinary perturbation theory, a non-perturbative analytic procedure and the Hermite spectral method for various classes of waves. Once its solutions are known, the overall amplitude factor may be found by using the wave action equation as shown in the text. When the method of multiple scales is invalid, as appears true of the tropical ocean, it is shown that Hermite spectral methods in latitude are much more accurate (at least in the absence of coastal boundaries) than the finite-difference methods used in the past. The techniques discussed here are applied to several classes of observed atmospheric equatorial waves in Part II (Boyd, 1978a).
Abstract
The analytical and numerical methodology of Boyd (1978) is applied to observed atmospheric waves. It is found that the structure and vertical wavelength of the stratospheric Kelvin wave of 15-day period and the tropospheric Kelvin wave of 40–50 day period are both negligibly affected by even the strongest shear. In contrast, the shear of the quasi-biennial oscillation can decrease the wavelength of the stratospheric n=0 mixed Rossby-gravity wave of 5-day period by 60% and produce changes of 50–100% in wave fluxes and velocities. The structure of synoptic-scale easterly waves (n=1 Rossby waves of 5-day period) is not drastically altered by shear, but the wavelength is tripled. This makes it unlikely that one can construct a quantitative wave-CISK theory of this mode without including latitudinal shear.
Abstract
The analytical and numerical methodology of Boyd (1978) is applied to observed atmospheric waves. It is found that the structure and vertical wavelength of the stratospheric Kelvin wave of 15-day period and the tropospheric Kelvin wave of 40–50 day period are both negligibly affected by even the strongest shear. In contrast, the shear of the quasi-biennial oscillation can decrease the wavelength of the stratospheric n=0 mixed Rossby-gravity wave of 5-day period by 60% and produce changes of 50–100% in wave fluxes and velocities. The structure of synoptic-scale easterly waves (n=1 Rossby waves of 5-day period) is not drastically altered by shear, but the wavelength is tripled. This makes it unlikely that one can construct a quantitative wave-CISK theory of this mode without including latitudinal shear.
