# Wave-Mean-Flow Interactions in Atmospheric Fluid Flows

Communicated by *Notices* Associate Editor Reza Malek-Madani

Geophysical fluid dynamics describes the flow of gases and liquids in the Earth’s atmosphere, oceans, and other large bodies of water such as lakes and rivers. From a mathematical perspective the evolution of the fluid flow can be described by partial differential equations derived from the physical principles of conservation of mass, momentum, and energy, and including terms that represent the effects of the rotation of the Earth and the effects of buoyancy and gravity. Analyses and solutions of these equations contribute to our understanding of the mechanisms that drive weather over the short term and climate over the long term.

Waves occur naturally in geophysical fluid flows, either in the interior of a body of fluid or at the interface between two fluids such as on the surface of the ocean. The primary mechanisms that generate waves in geophysical flows are the Coriolis force that arises from the Earth’s rotation and the buoyancy and gravitational forces. Wave interactions take place over a wide range of temporal and spatial scales and can have profound effects on the general circulation of the fluid flow. Thus, an understanding of wave properties, including their generation, propagation and interaction with the general circulation, is of great importance for accurate weather forecasting and climate modeling. Forecasting depends on frequent observations and measurements of the properties and current state of the atmosphere, analyses of the data obtained, a good understanding of the dynamics of the atmosphere, the development of mathematical models and computational algorithms, powerful computers, careful analysis and interpretation of the output of the models, and presentation of the output in the form of forecasts.

Mathematical analyses of the fluid equations provide insights that inform the development of the models. Waves may be represented in the equations as perturbations to a suitably averaged background fluid flow. This perspective leads to nonlinear perturbation equations for the wave quantities, which can be linearized and solved exactly in certain special configurations, and can be analyzed in more general situations using the methods of perturbation theory, hydrodynamic stability analysis, and asymptotic analysis. In general, the goal is to understand the *wave-mean-flow interactions*, i.e., the interchange of energy and momentum between the perturbations and the large-scale background mean flow.

This feature article discusses a mechanism by which certain types of waves interact with a background mean flow in a geophysical fluid context: the critical layer interaction. This type of interaction occurs when a wave propagating with a given phase speed reaches a location called a critical line or critical level, where the speed of the background mean flow is equal to the wave phase speed. In the critical layer surrounding this line, the wave transfers momentum and energy to the background fluid flow, a process called wave absorption, and modifies the flow as a result. This can cause reversal of wind direction, changes in the mean vorticity and temperature, wave breaking, and turbulence. Mathematically, the critical layer corresponds to a singularity in the linearized inviscid equations for steady-amplitude waves.

This article is not intended to be a comprehensive review of atmospheric waves and critical layer phenomena. The focus of the discussion shall be on three types of waves that are ubiquitous in the atmosphere, namely internal gravity waves, planetary Rossby waves, and vortex Rossby waves, and their corresponding critical layer theories. In spite of the differences between the physical characteristics of these waves, there are some similarities in the underlying mathematics and in the approaches used for analyses and for obtaining solutions. The article aims to highlight some of these similarities by referring to critical layer studies of the author and other researchers.

Gravity waves exist in geophysical fluid flows as a result of the competing effects of the downward force of gravity and the upward buoyancy force that causes fluid particles to rise. They include surface gravity waves, such as the familiar water waves that form at the interface between a body of water and the air above.

Internal gravity waves occur in the interior of a fluid, e.g., within the atmosphere or within the ocean, and can propagate vertically as well as horizontally. In the atmosphere, the most common examples are orographic waves that are generated by airflow over mountains and other large obstacles and convective waves that form above cloud layers and arise from vertical motion in the lower atmosphere. The spatial scale of these waves is small relative to the planetary scale; their horizontal wavelengths are typically no more than a few hundred kilometers.

Figure 1 shows an image of gravity waves over stratocumulus cloud layers above the Indian Ocean; they appear as multiple cloud bands corresponding to the troughs and crests of the waves. A similar structure is seen in the clouds in the photograph in Figure 2.

Internal gravity waves have frequent and profound effects on the general circulation of the atmosphere over different time scales and they thus affect both weather and climate. The amplitude of an internal gravity wave tends to increase as the wave propagates upwards. This is because the wave kinetic energy is proportional to the product of the atmospheric density and the square of the amplitude of the fluctuation in the velocity. Conservation of the wave energy in an atmosphere with density decreasing with altitude implies that the wave amplitude increases with altitude. In practice, the rate of increase could be modified by dissipation; this is seen, for example, in the measurements reported by Tsuda Tsu14. The amplitude increase can lead to an instability and possible wave breaking at high altitudes. Moreover, the presence of a critical level acts to filter out the waves, as observed by Whiteway and Duck WD96, and may also lead to breaking. Breaking waves generate the turbulence that is often experienced by air travelers especially when flying over mountain ranges or over convective clouds.

On a longer timescale, internal gravity waves propagating into the stratosphere transfer momentum and energy to the background flow and are known to contribute to the development of the quasi-biennial oscillation (QBO). This is a large-scale oscillation in which the wind direction alternates between eastward and westward phases with a period of 26–30 months. The QBO has profound effects on the stratospheric flow globally, beyond the tropics; it affects the transport and distribution of chemical constituents in the atmosphere and may even affect the strength of hurricanes B 01. Given the important role that atmospheric gravity waves are known to play in such weather and climate events, it is important to represent and simulate them accurately in the general circulation models that are used for weather prediction and climate modeling. Historically, this presented a challenge due to their relatively small wavelengths and short time periods. The development of the mathematical theory of gravity waves, including critical layer interactions, has contributed to our understanding of their effects on the general circulation and helped to improve the accuracy of their representation in general circulation models in recent decades FA03.

Rossby waves are oscillations that result from the variation of the rotational or Coriolis force due to the curvature of the Earth. Gravitational and buoyancy effects also play an albeit less significant role in their generation and propagation. These waves play a significant role in weather in the midlatitude regions and are associated with oscillations in the jet stream. Rossby waves are named after the Swedish-born meteorologist Carl-Gustaf Arvid Rossby, who first identified them in 1939 as large-scale wavy patterns in the atmospheric flow and subsequently described their characteristic features. The mathematical theory of Rossby wave critical layers dates back to the 1950s and 60s Lin55. The seminal mathematical studies on the nonlinear inviscid time-dependent problem, presented simultaneously by Stewartson Ste78 and Warn and Warn WW78 in 1978, describe asymptotic solutions that are commonly called the “SWW” solutions. The SWW theory provides insight into the observed formation of the so-called stratospheric “surf-zone,” a large region of planetary wave breaking that extends from northern latitudes towards the tropics MP84.

Rossby waves also exist in the atmosphere on a smaller scale as vortex waves, which take the form of outward-propagating disturbances within cyclonic vortices such as hurricanes Mac68, and result primarily from the force due to the radial gradient of the cyclone vorticity. A hurricane is a cyclone with maximum wind over 32 that develops in the atmosphere over the tropical ocean. Typically, a hurricane has a calm area with low atmospheric pressure at its centre. This is called the eye and it is surrounded by a ring of extremely high angular velocity called the primary eyewall. In high-intensity hurricanes, a secondary eyewall sometimes develops as a larger outer ring, increases in intensity and moves inward to replace the primary eyewall. This process repeats as an eyewall replacement cycle. The structure of the vortex at two stages in this process is shown in Figure 3.

Vortex Rossby waves interact with the background vortex flow if there is a critical radius where the vortex flow angular velocity is equal to the wave phase speed. The presence of a critical radius affects the structure and intensity of the hurricane and it has been suggested that vortex Rossby waves may contribute to the secondary eyewall development and the eyewall replacement cycle, possibly through a critical radius interaction mechanism MK97.

All these examples underscore the importance of characterizing and understanding atmospheric waves and their behavior at critical layers. This article discusses the critical layer theory for the three types of waves mentioned above. In each case, the discussion is on situations where the wave dynamics is represented in a two-dimensional spatial domain by perturbation equations which are analyzed using asymptotic methods or solved numerically. The focus is on problems involving forced waves, i.e., where an oscillatory boundary condition is applied at one end of the two-dimensional domain and generates a wave with a specified wavelength which propagates into the interior of the region where it encounters a critical layer.

In Section 1 the basic equations of fluid dynamics are presented and perturbation equations are obtained from these. Sections 2–4 give respective overviews of planetary Rossby waves, vortex Rossby waves, and internal gravity waves, and the aspects of their critical layer theories that are similar. Some results of recent numerical studies of the author are presented as illustrations.

## 1. Geophysical Fluid Flows and Waves

The dynamics of geophysical fluid flows can be described mathematically by equations based on the laws of conservation of mass, momentum, and energy. In a geophysical fluid configuration the flow exists in a three-dimensional domain defined by the spherical Earth, but depending on the spatial scale of the phenomena of interest, we may be able to justify using a two- or three-dimensional representation with Cartesian coordinates as an approximation for spherical geometry.

In terms of Cartesian coordinates , and , the conservation of mass is written in terms of the fluid density (mass per unit volume) , the components , , and , of the fluid velocity in the respective directions, and the time as the continuity equation ,

The special case of an incompressible flow, where the rate of change of the density along fluid parcel trajectories is considered to be zero, gives a simplified form of the continuity equation

The conservation of momentum is given by Newton’s Second Law, which says that the rate of change of the fluid momentum (the mass multiplied by the velocity) in some control volume of fluid is balanced by the sum of any applied forces. Considering a fluid parcel of unit volume moving with velocity components , and , this can be written as ,

The differential operator defined above, gives the rate of change of a fluid property following a fluid parcel moving with velocity , this is known as the Lagrangian derivative. ;

The terms on the right-hand side are the forces acting on the fluid: is the pressure (the force per unit area), the spatial derivatives of are the components of the pressure gradient force, , , are the components of the Coriolis force and is the acceleration due to gravity, which gives a gravitational force that acts downward and thus only appears in the vertical momentum equation. Other forces that could be present in a geophysical flow situation have been omitted here, for example the viscous or frictional force and any forces due to electromagnetic effects.

Given the current state of the fluid flow (at in some specified region in space, and some information about the flow at the boundaries, the goal is to solve the equations to find out its state at a later time (for ) within that region. This can be written as an initial-boundary-value problem for the system of partial differential equations. To facilitate mathematical analyses, and in particular, to assess the relative magnitude and importance of the various terms, the equations may be considered in nondimensional form. )

In a weakly-nonlinear analysis, waves in a fluid are considered as perturbations to a background mean flow with each fluid flow quantity being written as the sum of its mean and a perturbation which is taken to be of small amplitude relative to the mean flow. In a nondimensional framework, we write a quantity as

where

A wave may reach a location in the fluid flow where the background flow speed is equal to the wave phase speed. In an actual three-dimensional flow, this would, in general, be a surface oriented in three-dimensional space. In a two-dimensional representation, it is a *critical line*. For planetary Rossby waves propagating on a horizontal plane, it is a critical latitude; for vortex Rossby waves traveling radially outward from the center of a vortex, it is a critical radius; and for gravity waves in a vertical plane, it is a critical level.

In the linearized theory for steady-amplitude waves in a flow with no viscosity, the critical line corresponds to a singularity in the ordinary differential equation for the wave amplitude. However, the singular behavior is a mathematical artifact resulting from the omission of the nonlinear terms and viscous terms and the assumption that the wave amplitude is steady. The methods of asymptotic analysis are useful for obtaining an approximate “inner” solution in the critical layer, which is then matched with the solution in the “outer” region away from the singularity. The inner solution is obtained by restoring the viscous terms to the equations in the critical layer, by considering waves with time-dependent amplitude, or by carrying out a weakly-nonlinear analysis in which the previously neglected nonlinear terms are included but they are considered to be “small” relative to the linear terms. In the nonlinear time-dependent formulation, the methods of multiple-scale analysis are used as well; we introduce a suitably scaled “slow” time variable and obtain a slowly varying solution to complete the description of the solution at late time.

The focus of the discussion here will be on weakly-nonlinear time-dependent solutions of the inviscid equations. The development of nonlinear time-dependent critical layer theory for Rossby waves and internal gravity waves has an extensive history developed over the past 60 years. A brief overview of critical layer theory is given here, along with some illustrative numerical simulations for Rossby waves (Sections 2 and 3) and internal gravity waves (Section 4). Each of the problems discussed is based on a two-dimensional simplification of the full three-dimensional system of conservation laws. In each case, the geometry of the problem is defined as either a horizontal or a vertical plane, to give an approximate representation of the direction of propagation of the type of wave studied. Such two-dimensional studies provide us with insights into the dynamics of the respective types of waves in more realistic three-dimensional situations.

## 2. Planetary Rossby Waves

In this section, the two-dimensional configuration involving Rossby waves in a barotropic shear flow is discussed. A barotropic fluid flow is one in which the density is a function of the pressure only, so that the surfaces of constant density coincide with the surfaces of constant pressure. In situations where the wave propagation is primarily horizontal, a barotropic flow on a two-dimensional horizontal plane may be a reasonable first approximation for the full three-dimensional representation. On the horizontal plane the density is taken to be constant; thus, mass conservation is described by the incompressible form of the continuity equation.

We consider an inviscid barotropic fluid flow on a horizontal plane defined by Cartesian coordinates

The terms

The form of the incompressible continuity equation in two variables 4 allows us to define a streamfunction

Here the subscripts

The total streamfunction is written in the form suggested by 1. We write

where

If we consider

Different approaches may be taken to analyze this linear equation, depending on the specification of the background flow and domain and on the information that we wish to obtain about the solutions. In the case where the mean flow speed

Here

The ratio

which defines the direction of propagation of the wave energy. To illustrate the difference between the phase velocity and the group velocity, we may consider a wave packet like that shown in the diagram in Figure 5 propagating in the direction of a horizontal or zonal variable

In the general case with mean flow speed

where

This is a generalization of the Rayleigh equation, which was first introduced by Lord Rayleigh in the 1880s for the situation with

In the approach of hydrodynamic stability theory, equation 10 is considered in the context of an eigenvalue problem, where *stability* refers to the situation where

The main result obtained from the stability analysis is that if a wave perturbation is unstable, with an amplitude that grows exponentially with

The Rayleigh–Kuo equation also describes the behavior of forced Rossby waves near a critical line. In the forced wave problem, an oscillatory boundary condition with a specified wave speed

where the argument

We consider the forced wave problem in which the solutions of 7 are in the form of waves with time-dependent amplitude

This leads to an initial-boundary-value problem for a partial differential equation involving

Stewartson Ste78 and Warn and Warn WW78 continued the time-dependent investigations by considering the nonlinear equation 6. The nonlinear “SWW” solution, as it is commonly called, behaves like the linear solution at early time in the sense that there is a reduction in the wave amplitude across the critical layer. This reduction is balanced by a transfer of momentum to the mean flow indicating that the wave is indeed absorbed by the mean flow, as the linear solutions suggest. In the nonlinear formulation, the wave absorption results in a wave-induced mean flow acceleration in the vicinity of the critical line. An equation that describes how the mean flow velocity changes in time as a result of the wave interaction is obtained by taking the

The “over-bar” on the right-hand side denotes the

Over time, additional nonlinear features develop in the critical layer. At late time momentum and energy from the mean flow are transferred to the wave modes as the wave is “reflected” by the mean flow. The timescale on which these nonlinear effects develop depends on the parameter

Contour plots of the total vorticity, calculated from the asymptotic solutions Ste78WW78 and from numerical solutions Bél76Cam04, show closed contours known as “cat’s eyes” in the critical layer, that eventually overturn and break. Figure 6 shows a contour plot of overturning vorticity contours in a nonlinear Rossby wave critical layer. This result was obtained from a time-dependent numerical solution of 6 in a rectangular domain given by nondimensional variables

The change in the mean flow velocity is computed numerically as

## 3. Vortex Rossby Waves in a Tropical Cyclone

Vortex Rossby waves also exhibit critical layer interactions which can modify the background flow. Although they differ greatly from planetary Rossby waves with respect to their scale, their structure, and the mechanisms that generate them, there are nevertheless strong qualitative similarities between their critical layer behavior and that of planetary Rossby waves. The development of vortex wave critical layer theory is more recent; in particular, in the past 25 years there have been a number of investigations into the linear initial value problem on a horizontal plane, with constant Coriolis parameter, e.g., BM02MK97. More recently, we NC15aNC15b examined a forced wave problem in a configuration analogous to the SWW problem, where the waves are forced by a oscillatory boundary condition at one end of the domain and propagate towards a critical layer.

In our configuration, we consider a background vortex flow with steady uniform rotation in an annular domain given by

The wave propagates radially outwards until it reaches a critical radius,

The quadratic profile considered for the mean flow angular velocity gives steady-amplitude solutions of the linearized perturbation equation in terms of hypergeometric functions. These solutions show behavior that is similar to that of planetary Rossby waves in the analogous steady-amplitude configuration Lin55: the amplitude of the perturbation is attenuated at the critical radius and there is a logarithmic phase change of

A weakly-nonlinear analysis of the nonlinear time-dependent problem, with the beta-effect included, reveals certain aspects of the solutions that bear resemblance to the observed features of the secondary eyewall replacement cycle in tropical cyclones NC15b. In particular, there is an inward displacement of the location of the critical radius with time and additional rings of high wave activity develop corresponding to the critical radii of the components of the solution that arise from the beta-effect.

Some numerical solutions for this configuration are shown in Figure 8: the wave streamfunction

## 4. Internal Gravity Wave Packet Generated by an Isolated Mountain

In this section we give an overview of linear and nonlinear critical layer problems for internal gravity waves in a two-dimensional configuration. Internal gravity waves affect the general circulation of the atmosphere through critical level interactions and through wave breaking and saturation. Gravity wave saturation occurs when a wave propagating upwards in the atmosphere reaches a level where its amplitude becomes too large to be sustained and the wave breaks partially to reduce its amplitude and maintain stability. Given the relatively short wavelengths of gravity waves, historically, general circulation models have had difficulty in resolving them adequately and correctly representing the drag force that results from gravity wave critical layer and saturation effects. Consequently, the need to develop and understand the mathematical theory of atmospheric gravity waves has been recognized for many decades.

Gravity wave critical layer theory, in particular, has a rich history that dates back to the 1950s and which developed in parallel to that of planetary Rossby waves. Because they propagate vertically as well as horizontally, the simplest two-dimensional model for gravity waves would be in a vertical plane with one horizontal coordinate tangent to the curved surface of the Earth and one vertical coordinate, normal to the tangent plane.

To a first approximation, the viscous and heat-conduction terms in the momentum and energy conservation equations may be omitted, but just as in the Rossby wave problems, in order to deal with singular solutions and secular terms in the critical layer context, we need to restore these terms to the equations, at least in the inner region, or alternatively consider a time-dependent linear or nonlinear formulation. As in sections 2 and 3, the discussion here will focus on situations with the viscous or heat-conduction terms omitted.

In addition, the Boussinesq approximation is often used to simplify the equations, making them more tractable to mathematical analyses and solutions, while still representing the vertical stratification of the fluid density and temperature. In this approximation, the fluid density is replaced by a background density

This is analogous to 4, but in terms of one horizontal variable and one vertical variable, rather than two horizontal variables. Again, this two-dimensional continuity equation allows us to define a streamfunction, but in this case it is given by

Differentiating the

Here the Laplacian

where

Neglecting the terms involving

The constant

and gives a measure of the extent of stratification of the density.

The problems defined by equations 15–17 share some similar features with the corresponding Rossby wave problems described in Section 2. All the approaches for analysis described in Section 2 can be applied here as well. If we consider both the mean flow speed

where the horizontal and vertical wavenumbers are denoted, respectively, by

Also, in analogy with the Rossby wave problem, the derivatives of the dispersion relation with respect to

In the general case with mean flow speed

where

The amplitude function

which is the analog of the Rossby wave Rayleigh–Kuo equation 10 for gravity waves.

As noted in Section 2, the Rayleigh–Kuo theorem gives us a necessary condition for stability of normal mode solutions 9 of 10. The analogous stability theorem for the gravity wave configuration gives a condition in terms of the Richardson number of the background flow. The Richardson number is a function of