Linear Gravity Waves Research Articles

Diurnal Off-Mountain Propagation of Rainfall and Low-Level Vertical Velocity over the Lee Side of the Yungui Plateau

Abstract Diurnal off-mountain propagation is a distinctive feature of rainfall over terrestrial areas, whereas the causes of this phenomenon are not well understood. Focused on the rainfall downstream of the Yungui Plateau (YGP), this study aims to examine whether the gravity waves stimulated by an assumed terrain-related thermal forcing could explain the feature. The results show that the rainfall diurnal phase propagates eastward at a speed of approximately 13 m s−1 over the lee side of YGP during warm seasons. The diurnal amplitude reaches its zonally maximum over the slope of the YGP and drops sharply downstream the terrain. The low-level vertical velocity exhibits similar diurnal characteristics. A linear model forced by a hollow heating is proposed to mimic the thermal forcing related to a mountain. Experiments with the model show that there are mainly two branches of waves around the terrain. One is over the upstream with upwind-tilting phase lines that move toward −z and −x directions. The other branch exists over the lee side of terrain with downwind-tilting phase lines that move toward −z and +x directions, which is considered to be relevant to the off-mountain propagation feature. The wave behavior over the YGP is then reproduced using the model. It is shown that the main features of the diurnal phase lag and the zonal amplitude distribution pattern of the low-level updraft could be captured by the model, suggesting an important role of the gravity wave in driving the diurnal propagation of vertical velocity and rainfall downstream large terrains. Significance Statement The purpose of this study is to better understand why diurnal off-mountain rainfall propagation exists over the downstream of Yungui Plateau (YGP). This is important because the off-mountain rainfall propagation influences the rainfall events over the lee side of mountains and can further influence people’s lives therein. Inspired by previous studies on coastal rainfall, we proposed a simplified linear gravity wave model forced by a topography-related heating function. The results showed that the thermally forced gravity waves could reproduce the main features of the propagating phase and amplitude pattern of upward motion disturbance over the lee side of the YGP. Our results highlight the importance of gravity waves stimulated by topography-related heating on the diurnal propagation features over the downstream of mountains.

Rainfall Enhancement Downwind of Hills Due to Stationary Waves on the Melting Level and the Extreme Rainfall of December 2015 in the Lake District of Northwest England

This paper investigates how stationary gravity waves generated by flow over orography enhance rainfall, with particular attention to the role of induced waves in the melting level. The findings reveal a new mechanism by which gravity wave flow focuses precipitation, amplifying rainfall intensity downwind of hills. This mechanism, which depends on the differential velocities of rain and snow, offers fresh insights into how orographic effects can intensify rainfall. A two-dimensional diagnostic model based on linear gravity wave theory is used to investigate the record-breaking rainfall of December 2015 in the Lake District of northwest England. The pattern of ascent is shown to have a qualitatively good fit to that of the Met Office’s operational high-resolution UKV model averaged over 24 h, suggesting that orographically excited stationary waves were the principal cause of the rain. Precipitation trajectories imply that a persistent downstream elevated wave caused by the Isle of Man supported a spray of seeding ice particles directed towards the Lake District, and that these grew whilst suspended in strong upslope flow before being focused by the undulating melting-level into intense shafts of rain.

General solution for linear 2D anisotropic surface gravity water waves on uniform current

Measuring and modeling ocean waves is a routine task for oceanographers and marine engineers. Accounting for current effects on waves is less common; however, such effects can be non-trivial, particularly for strong currents. Recent research has also found that currents are increasing with global temperatures. Thus, accounting for currents is of potential importance to a growing subset of real-world environments. One significant impact of currents is their modification of dispersion. It is known that, in contrast to dispersion without current, wave–current dispersion is multivalued and anisotropic, meaning it varies with direction and can yield multiple solutions for the same frequency. Strictly speaking, a complete analysis ought to account for all dispersion solutions. This study presents a general wave field solution for linear steady-state two-dimensional surface gravity waves on current. In contrast to existing formulations, it accounts for the complete set of dispersion solutions. The multiple solutions correspond to linearly independent (same frequency) spatial modes with no analogy in models without current. To fully determine the complete temporal and spatial features of a wave field, one must determine the amplitudes of all spatial modes. This can be achieved through the solution of an inverse problem, provided one has sufficient initial conditions. The significance of accounting for all spatial modes is demonstrated through examples. It is shown that a single time-series measurement is insufficient for determining the spatial features of a monochromatic (single frequency) wave. Rather, additional spatial information must be introduced to render the problem well-posed. The theory and methodology presented can be used to improve wave models and measurements.

Low-frequency Internal Gravity Waves Are Pseudo-incompressible

Starting from the fully compressible fluid equations in a plane-parallel atmosphere, we demonstrate that linear internal gravity waves are naturally pseudo-incompressible in the limit that the wave frequency ω is much less than that of surface gravity waves, i.e., ω≪gkh , where g is the gravitational acceleration and k h is the horizontal wavenumber. We accomplish this by performing a formal expansion of the wave functions and the local dispersion relation in terms of a dimensionless frequency ε=ω/gkh . Further, we show that, in this same low-frequency limit, several forms of the anelastic approximation, including the Lantz–Braginsky–Roberts formulation, poorly reproduce the correct behavior of internal gravity waves. The pseudo-incompressible approximation is achieved by assuming that Eulerian fluctuations of the pressure are small in the continuity equation—whereas, in the anelastic approximation, Eulerian density fluctuations are ignored. In an adiabatic stratification, such as occurs in a convection zone, the two approximations become identical. However, in a stable stratification, the differences between the two approximations are stark and only the pseudo-incompressible approximation remains valid.

Semi-Implicit Computation of Fast Modes in a Scheme Integrating Slow Modes by a Leapfrog Method Based on a Selective Implicit Time Filter

Abstract A scheme for integration of atmospheric equations containing terms with differing time scales is developed. The method employs a filtered leapfrog scheme utilizing a fourth-order implicit time filter with one function evaluation per time step to compute slow-propagating phenomena such as advection and rotation. The terms involving fast-propagating modes are handled implicitly with an unconditionally stable method that permits application of larger time steps and faster computations compared to fully explicit treatment. Implementation using explicit and recurrent formulation is provided. Stability analysis demonstrates that the method is conditionally stable for any combination of frequencies involved in the slow and fast terms as they approach the origin. The implicit filter used in the method damps the computational modes without noticeably sacrificing the accuracy of the physical mode. The O[(Δt4)] accuracy for amplitude errors achieved by the implicitly filtered leapfrog is preserved in applications where terms responsible for fast propagation are integrated with a semi-implicit method. Detailed formulation of the method for soundproof nonhydrostatic anelastic equations is provided. Procedures for implementation in global spectral shallow-water models are also given. Examples comparing numerical and analytical solutions for linear gravity waves demonstrate the accuracy of the scheme. The performance is also shown in more practical nonlinear applications, where numerical solutions accomplished by the method are evaluated against those computed from a scheme where the slow terms are handled by the third-order Runge–Kutta scheme. It demonstrates that the method is able to accurately resolve fine-scale dynamics of Kelvin–Helmholtz shear instabilities, the evolution of density current, and nonlinear drifts of twin tropical cyclones.

A simple approximation for the drift rates of rotating polygons on a free fluid surface

We report on laboratory experiments investigating the dynamics of a free fluid surface in a cylindrical tank with a rotating bottom plate. The shear instability in the system creates polygonal structures, which propagate around the domain, along the fixed vertical sidewalls. We analyze the wavelengths and drift rates of these patterns, which are known from previous literature to be created by a complex interplay between centrifugal effects and gravity wave propagation in this unique geometry. We find that with an empirical correction factor, the drift rates of the polygonal vortices can be approximated fairly well by a surprisingly simple formula derived from the dispersion relation of linear gravity waves using easily observable parameters.

Wave Boundary Layer at the Ice–Water Interface

On re-examining the problem of linear gravity waves in two layers of fluids with a viscous ice layer overlaying water of deep depth, we give a detailed analysis of the fluid velocities, velocity shear, and Reynolds stress associated with wave fluctuations in both the ice layer and the wave boundary layer just beneath it. For the turbulent wave boundary layer, water eddy viscosity is used. Comprehensive discussions on various aspects of the velocity fields are made in terms of a Reynolds number based on the ice-layer thickness and viscosity, and the ice-to-water viscosity ratio. Speculation of the wave-induced steady streaming is made based on the Reynolds stress distribution, offering a preliminary insight into the mean flows in both the ice layer and wave boundary layer in the water. For wave attenuation, the results using a typical ice viscosity and a reasonable water eddy viscosity show good agreement with data over the range of frequencies for field and lab waves, significantly outperforming those assuming an inviscid water.

Assessment of a porous viscoelastic model for wave attenuation in ice-covered seas

Chen et al. (2019) recently proposed a two-dimensional continuum model for linear gravity waves propagating in ice-covered seas. It is based on a two-layer formulation where the ice cover is viewed as a porous viscoelastic medium. In the present paper, extensive tests against both laboratory experiments and field observations are performed to assess this model’s ability at describing wave attenuation in various types of sea ice. The theoretical predictions are fitted to data on attenuation rate via error minimization and numerical solution of the corresponding dispersion relation. Detailed comparison with other existing viscoelastic theories is also presented. Estimates for effective rheological parameters such as shear modulus and kinematic viscosity are obtained from the fits and are found to vary significantly among the models. For this poroelastic system, the range of estimated values turns out to be relatively narrow in orders of magnitude over all the cases considered. Against field measurements from the Arctic Ocean, this model is able to reasonably reproduce the roll-over of attenuation rate as a function of frequency. Given the rather large number of physical parameters in such a formulation, a sensitivity analysis is also conducted to gauge the relevance of a representative set of them to the attenuation process.

Identification of acoustic-gravity waves from satellite measurements

A method for recognizing the types of linear acoustic gravity waves (AGWs) in the atmosphere from satellite measurements is proposed. It is shown that the polarization relations between fluctuations of wave parameters (velocity, density, temperature, and pressure) for freely propagating waves, as well as evanescent wave modes, differ significantly, which makes it possible to identify different types of atmospheric waves in experimental data. A diagnostic diagram is proposed, with the help of which, from the phase shifts of the observed parameters, it is possible to determine the type of wave, as well as the direction of its movement relative to the vertical. Using phase shifts between fluctuations of the velocity and thermodynamic parameters of the atmosphere, not only the type of the wave, but also its spectral characteristics can be determined. The verification of the proposed method for identification of polar wave disturbances in measurements from the low-orbit satellite Dynamics Explorer 2 was carried out. The verification showed that the polarization relations of AGW in the thermosphere mainly correspond to the gravity branch of acoustic-gravity waves freely propagating in the direction from bottom to top. This conclusion is consistent with other results of AGW observations in the atmosphere and ionosphere by ground-based and satellite methods. No evanescent waves were observed on the considered orbits.

Effects of a Hole on Uplifting Forces on a Submerged Horizontal Thin Plate

The effect of a hole on uplifting forces acting on a submerged horizontal thin plate are examined using the boundary element method. To deal with the submerged thin plate case, which is a degenerate boundary problem, a hypersingular integral equation is employed to overcome some troubles when the boundary element method is used. Two-dimensional problems are considered, and the linear gravity wave theory is adopted. Several examples are calculated and the numerical results are presented to show the influence of a hole on uplifting forces.

Effects of a Hole on Uplifting Forces on a Submerged Horizontal Thin Plate

The effect of a hole on uplifting forces acting on a submerged horizontal thin plate are examined using the boundary element method. To deal with the submerged thin plate case, which is a degenerate boundary problem, a hypersingular integral equation is employed to overcome some troubles when the boundary element method is used. Two-dimensional problems are considered, and the linear gravity wave theory is adopted. Several examples are calculated and the numerical results are presented to show the influence of a hole on uplifting forces.

Surface gravity wave interaction with a partial porous breakwater in a two-layer ocean having bottom undulations

Linear surface gravity wave scattering by a partial porous breakwater is studied in a two-layer ocean having an interface with bottom undulations. The partial breakwaters, namely surface-piercing breakwater (SPB) and bottom-standing breakwater (BSB), are considered. The continuity of fluid pressure and mass flux is applied at interface boundaries, while mass conserving conditions are employed to account for slope discontinuity in seabed geometry. Porous breakwaters' reflective and dissipative characteristics are studied under wave-wave interaction. The study reveals that when a mound bottom is present, more incident waves in interface mode are transmitted through BSB, while more waves in surface mode are dissipated. More incident waves in surface mode are transmitted through SPB, while more waves in interface mode are dissipated. In contrast, more transmission occurs in surface mode through both BSB and SPB for a slope-shelf bottom. However, the breakwater dissipates a substantial amount of incident-wave energy in interface mode. Further, a considerable reduction in wave transmission occurs due to higher reflection by the interface-piercing porous breakwaters, and the breakwaters experience high wave loads. Relatively, porous breakwater dissipates less wave energy in shallow depth. The study reveals that the dead water effect is weakened by the porous breakwater.

A single-mode approximation for gravity waves in the thermosphere

In previous work (Knight et al., 2019) a multilayer Fourier method was introduced for obtaining solutions for a partial differential equation (PDE) associated with a dispersion relation for linear gravity waves in a vertically varying, anelastic, nonhydrostatic, viscous, and thermally diffusive atmosphere over some altitude range, given boundary conditions that allow waves to enter at the lower boundary and exit at the top. The previous work described an exact solution method for the dispersion-relation PDE using all wave modes. The current work describes a type of approximate solution, based on a single upgoing gravity-wave mode for each Fourier component, which is computationally cheaper and simpler to implement. Two different approaches to defining the single-mode solution are presented, one based on the Wentzel–Kramers–Brillouin (WKB) method and the other based on a simplified version of the multilayer approximation. The two approaches are shown to be mathematically equivalent. The accuracy of the single-mode approximation is investigated through comparison with the all-mode solution for a realistic example in which gravity waves generated by a convective storm in the troposphere propagate into the upper thermosphere and for which the single-mode approximation gives reasonably accurate results. It is necessary to include imaginary frequency shifts in some Fourier components in order for the different types of modes (e.g., upgoing vs. downgoing and gravity-wave vs. dissipative) to be well-defined in the presence of molecular viscosity. A systematic approach to determining the smallest possible imaginary frequency shift for any given Fourier component is introduced. The new method efficiently suppresses a specific type of numerical artifact associated with large imaginary frequency shifts.

The role of viscoelastic foundation on flexural gravity wave blocking in shallow water

A hydroelastic model is developed to study the interaction of linear long gravity waves with a very large floating flexible plate resting on a viscoelastic foundation, which is based on the Kelvin–Voigt model. Flexural gravity wave blocking occurs for specific values of the compressive force in the absence of viscous damping. During wave blocking, the group velocity vanishes, and mode swapping occurs. Within wave blocking and plate buckling limit in the presence of compressive force, three distinct propagating modes occur in the absence of viscous damping. Moreover, the study reveals that irrespective of the values of viscous damping constant, the blocking/buckling points shift to a higher wavenumber with an increase in the value of elastic foundation constant. On the other hand, the flexural gravity wave modes become complex in the presence of a viscoelastic foundation. The complex wave modes are classified as predominant progressive wave modes and rapidly decaying modes. In the presence of viscous damping, wave blocking does not happen before the buckling limit of the compressive force. However, the phase velocity vanishes, and the group velocity becomes continuous irrespective of the value of non-zero viscous damping at the buckling limit for the compressive force. The detailed behavior of the roots of the dispersion equation and the mode shapes are illustrated through contour plots and by analyzing the roots’ loci. Furthermore, plate deflections are exhibited for different wave and structural parameters.

Short-time deterministic prediction of individual waves based on space-time X-band Marine radar measurements

We demonstrate and verify, by the use of both synthetic and real wave data, a newly developed capability of short-time phase-resolved wave prediction based on incoherent X-band marine radar measurements. An inversion algorithm is developed to convert X-band radar sea surface measurements into the phase-resolved wave field and the associated wave spectrum based on the linear gravity wave theory. The wave components obtained from the reconstruction are then used to initialize the wave propagation model that is used to provide a short-time deterministic forecast of wave field evolution downstream. Both wave spectrum and spatial-temporal wave elevation evolution obtained based on the X-band measurements are compared with the independent point wave measurements by Miros RangeFinder. The agreements between them are reasonably well, which has a significant implication on practical applications of short-time deterministic wave prediction in optimal marine operations.

On the Dynamics of Atmospheric Bores

AbstractThe dynamics of a prototypical atmospheric bore are investigated through a series of two-dimensional numerical simulations and linear theory. These simulations demonstrate that the bore dynamics are inherently finite amplitude. Although the environment supports linear trapped waves, the supported waves propagate in roughly the opposite direction to that of the bore. Qualitative analysis of the Scorer parameter can therefore give misleading indications of the potential for wave trapping, and linear internal gravity wave dynamics do not govern the behavior of the bore. The presence of a layer of enhanced static stability below a deep layer of lower stability, as would be created by a nocturnal inversion, was not necessary for the development of a bore. The key environmental factor allowing bore propagation was the presence of a low-level jet directed opposite to the movement of the bore. Significant turbulence developed in the layer between the jet maximum and the surface, which reduced the low-level static stability behind the bore. Given the essential role of jets and thereby strong environmental wind shear, and given that idealized bores may persist in environments in which the static stability is constant with height, shallow-water dynamics do not appear to be quantitatively applicable to atmospheric bores propagating against low-level jets, although there are qualitative analogies.

Linear surface gravity waves on current for a general inertial viewer

Marine measurement instrumentation, such as free-floating wave buoys, drones, and autonomous unmanned vehicles, often propagates in different directions and velocities relative to the fluid and waves. Convention assumes that these different instrumentations provide Galilean invariant descriptions of the wave field. Herein, it is shown that Galilean invariance exists for the water wave problem only in a restricted sense. The impact of this loss of invariance is investigated using a new formulation of the water wave problem, which is generalized for both current and an arbitrary inertial viewer. In the still water limit, the boundary value problem is shown to be non-invariant under Galilean transformations. This impacts the dispersion relation and interpretation of measurements. It also explains the appearance of wave modes on current, which have no analogy on still water. These modes do not appear in a still water formulation because it is a degenerate representation exhibiting a loss of Galilean symmetries. The approach provides a more complete solution of the wave–current boundary value problem by making a clear distinction between current and viewer velocity effects. Numerical examples that demonstrate the importance of the results on calculating wave characteristics are given.

On the Spectra of Gravity Waves Generated by Two Typical Convective Systems

Spectral characteristics of lower-stratospheric gravity waves generated in idealized mei-yu front and tropical cyclone (TC) are compared by performing high-resolution simulations. The results suggest that the systems which organize convection in different forms can generate waves with distinctly different presentation. The mei-yu front appears as a linear zonal wave source and gravity waves are dominated by cross-frontal (meridional) propagating components. The northward (southward) components have dominant meridional wavelengths of 125–333 km (>250 km), periods of 100–200 min (83–143 min), and phase speeds of 0–15 m s−1 (15–20 m s−1). The TC appears as a point wave source and gravity waves propagate equally in various horizontal directions. The waves exhibit greater power and broader spectral distributions compared with those in the mei-yu front, with dominant horizontal wavelengths longer than 62.5 km, periods of 33–600 min, and phase speeds slower than ~40 m s−1.

Dispersion and attenuation in a porous viscoelastic model for gravity waves on an ice-covered ocean

Abstract A two-dimensional continuum model is proposed for linear gravity waves propagating across ice-covered seas. It is based on a two-layer formulation where the floating sea ice is described as a homogeneous isotropic poroelastic material and the underlying ocean is viewed as a weakly compressible fluid. Dissipative effects are taken into account by including viscosity in rheological properties of the ice layer. An exact dispersion relation is derived for traveling wave solutions of this coupled system and numerical estimates are obtained for its complex roots. Extensive tests are conducted to examine the dependence of results on various parameters in both the porous and non-porous cases. Detailed comparison with existing viscoelastic models is provided, and good agreement on both wave dispersion and attenuation is found. In the porous case with friction, a non-monotonic behavior is observed for the attenuation rate as a function of frequency, which is reminiscent of the roll-over phenomenon that has been reported in field observations.

Mixed Rossby–Gravity Wave–Wave Interactions

AbstractMixed triad wave–wave interactions between Rossby and gravity waves are analytically derived using the kinetic equation for models of different complexity. Two examples are considered: initially vanishing linear gravity wave energy in the presence of a fully developed Rossby wave field and the reversed case of initially vanishing linear Rossby wave energy in the presence of a realistic gravity wave field. The kinetic equation in both cases is numerically evaluated, for which energy is conserved within numerical precision. The results are validated by a corresponding ensemble of numerical model simulations supporting the validity of the weak-interaction assumption necessary to derive the kinetic equation. Since they are generated by nonresonant interactions only, the energy transfers toward the respective linear wave mode with vanishing energy are small in both cases. The total generation of energy of the linear gravity wave mode in the first case scales to leading order as the square of the Rossby number in agreement with independent estimates from laboratory experiments, although a part of the linear gravity wave mode is slaved to the Rossby wave mode without wavelike temporal behavior.