## 1. Introduction

Sound waves are generated from vortical flows (aeroacoustics). After the pioneering work of Lighthill [1], many theoretical and numerical works regarding this subject have been done. There are several good review papers and text books, for example, see [2–5].

Inertia‐gravity waves, in which buoyancy and Coriolis force provide the restoring force, are important in the atmosphere and ocean, because they drive general circulation in the middle atmosphere [6] and contribute to the ocean energy budget [7, 8]. Traditionally, rotating shallow water system has been used to study nonlinear interactions between vortex and wave [9] because this system is the simplest system in which both vortical flows and inertia‐gravity waves can exist. One of the typical examples is the Rossby adjustment process [10, 11], in which initial unbalanced state is assumed. Then inertia‐gravity waves (hereafter inertia‐gravity waves are referred to gravity waves) are radiated from unbalanced state toward balanced state. However, there are few works regarding the generation of gravity waves by unsteady motions of nearly balanced vortical flows in rotating shallow water system.

Ford's pioneering work [12] has shown that gravity waves are radiated from unsteady vortical flows. This type of gravity wave radiation is referred to as “spontaneous emission” [13], because initial balanced flows radiate gravity waves spontaneously during the time evolution. Since a shallow water system is equivalent to a two‐dimensional adiabatic gas system if the effect of Earth's rotation is negligibly small, gravity waves are analogous to sound waves. Using the acoustic analogy of Lighthill [1], Ford [12, 13] introduced a source of gravity waves. For the purpose of practical motivation, this new paradigm of spontaneous emission is intensively investigated; for example, see [14] and references therein. Recently, the theory of generation mechanism has been proposed [15, 16]. While it is pointed out that spontaneous emission in the shallow water system is different from that in the continuous stratified system [17], fundamental works from a viewpoint of geophysical fluid dynamics are nevertheless important [18].

As an extension of Ford's works [12, 13], several numerical works are performed in shallow water system on an *f*‐plane [19, 20] and a sphere [21]. While it is widely known that the effect of the Earth's rotation suppresses inertia‐gravity wave radiation, previous studies [20, 21] have reported that the effect of the Earth's rotation intensify gravity wave radiation in some parameter space. In this chapter, recent results of the inertia‐gravity wave radiation from nearly balanced vortical flows as an extension of sound wave generation from vortical flows are reviewed. Inertia‐gravity wave radiation from various types of vortical flows, such as a corotating vortex pair [22], elliptical vortex (Kirchhoff vortex) [23] and merging of (equal or unequal) vortices [24, 25], are investigated in a wide range of parameter space. All these works have reported that cyclone‐anticyclone asymmetry in spontaneous emission and a local maximum of intensity of gravity waves emitted from anticyclones at intermediate value of the Coriolis parameter *f*.

This chapter is organized as follows. In Section 2, the analytical derivation of the far fields of gravity waves is introduced for the cases of a corotating point vortex pair and an almost circular Kirchhoff vortex. The derived forms are verified quite well by the numerical simulation (Section 3). In addition, the results of gravity wave radiation from the merging of (equal or unequal) vortices are also introduced. Section 4 gives brief summary points and future issues.

## 2. Analytical estimate

In this section, the analytical derivation of the far fields of gravity waves from vortical flows is introduced. The derived form includes the effect of Earth's rotation in the source term, which causes the cyclone‐anticyclone asymmetry. Two examples of the corotating point vortex pair and Kirchhoff vortex are shown. See Refs. [22] and [23] for details.

### 2.1. Basic equation

Basic equations are the shallow water equations on an *f*‐plane, written as

where *x* and *y* directions in the Cartesian coordinates, respectively. The total depth of the fluid *f* and *g*, respectively.

Eqs. (1)–(3) are the same as for vortex sound (aeroacoustics) if the effect of the Earth's rotation is negligibly small. From Eqs. (1)–(3), Lighthill‐Ford equation is obtained [1, 12]:

where

where *Fr* (where *Fr* corresponds to Mach number (the ratio of the flow velocity to the phase speed of sound waves) in the field of aeroacoustics.

### 2.2. Source and far field

In the source

for nondivergent flow (

For the rotating case, on the other hand, the second term on the right‐hand side of Eq. (5) becomes also important for relatively larger *f* [20, 21]. Then, for the relevant term on the right‐hand side of Eq. (4), the following approximation holds

for nondivergent flow (

The Green's function of Eq. (4) in the two‐dimensional domain incorporating time variation is defined from the Klein‐Gordon equation:

where

where

### 2.3. Typical examples

To solve Eq. (10) analytically, two cases of a corotating point vortex pair and an almost circular Kirchhoff vortex are introduced as examples. **Figure 1** shows the schematics of these experimental configurations.

A point vortex pair with the same sign and strength corotates. For aeroacoustics, analytical [4] and numerical [26] studies are performed in this configuration. Zeitlin [9] also derived an analytical solution in nonrotating shallow water. A vortex pair with a circulation Γ positioned at distance 2*l* corotates at an angular velocity

Then the vorticity

where *z* direction and

From Eq. (6) the source associated with vortical flows is equivalent to the two‐dimensional quadrupole

where

Similarly, from Eq. (7) the source originating in the Coriolis acceleration is expressed as

As for the case of the Kirchhoff vortex, the derivation is in the same line as a corotating point vortex pair. The Kirchhoff vortex has a patch of constant vorticity,

where the semimajor axis of the ellipse is

Then, to first order in

After substituting Eqs. (15) and (17) in Eq. (10), the delta function can be integrated

Recalling that the following approximation in the far field (

then it follows from Eq. (21):

(23) |

With the form of Green's function Eq. (9), Eq. (23) for a corotating vortex pair is written as

where *B*, can be calculated by changing variables, *B* is expressed as

where the sign function

because

(27) |

where

(28) |

Eqs. (27) and (28) are applicable for both cyclone (*f*. In the absence of the Earth's rotation for *f*. First, spontaneous emission is suppressed for large *f* because of a small value in the square root and second parentheses. Second, the source Eq. (7) originating in the Coriolis acceleration acts oppositely to the gravity wave radiation caused by the second term in the first parentheses. Meanwhile, the source Eq. (7) originating in the Coriolis acceleration cancels out the source Eq. (6) associated with vortex, since the same signs of *f* for cyclone. In contrast, those two sources magnify each other for anticyclone. Then, gravity waves are intensely radiated from anticyclone. Simple explanations for the suppression of gravity wave radiation at large *f* are reported [9]. Note also that it is possible to derive analytical estimate in the case of evanescent gravity waves for

Examples of the far field of gravity waves (**Figure 2**. Here, the Rossby number (*b* in order to keep *Ro*, *F*r and the aspect ratio. Anticyclone radiates gravity waves more intensely than cyclone at relatively large *f* (**Figure 2a** and **b**) and there is no cyclone‐anticyclone asymmetry in spontaneous emission for the nonrotating case of **Figure 2c**).

The intensity of gravity waves for both cases of the corotating vortex pair,

to estimate the dependence on *r* in the field. **Figure 3** shows *b* cases (*Ro*~12.5, *f*~2/25) for the corotating point vortex pair and *R*o~1, *f*~1/10) for the Kirchhoff vortex. The cyclone‐anticyclone asymmetry is similar to both cases, though vortical flows are completely different. Note that *R*o is different among the vortices for the same value of *f* because the velocity and length scales depend on the vortical configuration.

## 3. Numerical simulation

To verify the analytical solution, a numerical simulation with a newly developed spectral method in an unbounded domain has been performed. The numerical results are in excellent agreement with the analytical results of Eqs. (27) and (28) [22, 23]. Furthermore, additional numerical simulations have been performed for the cases of merging of (equal or unequal) vortices, in which analytical solution cannot be derived [24, 25]. In this section, the results of numerical simulation as well as model settings are introduced.

### 3.1. Model settings

Shallow water equations on an *f*‐plane in polar coordinates are used for the numerical simulation. The equations of relative vorticity

where *u* and *v* are the velocities in the azimuthal (

where

Eqs. (31)–(33) are solved by a conformal mapping from a sphere *R* to a plane **Figure 4** shows a schematic of this mapping. With the aid of the following relation,

the transformation of the coordinates is expressed as

Then the phenomena on a two‐dimensional unbounded plane can be calculated on a sphere with an ordinary spectral method of spherical harmonics by this mapping. Since grid points are arranged nonuniformly (many grid points are positioned in the near field of vortical flows, while few are in the far field of gravity waves), this method enables us to simulate nonlinear interactions between vortical flows and gravity waves with high accuracy [22–25].

Additional advantage is that the term of the hyper viscosity, which is intended to dissipate unresolved small scales numerically in the spectral model, acts as a sponge layer. By the conformal mapping, the usual form of hyper viscosity can be written as

where *p* and

### 3.2. Verification

As an initial state, a corotating Gaussian vortex pair is used to mimic a point vortices, which is expressed as

where **Figure 1a**). The values of *Ro* is reciprocal of *f*. The field of

Similarly, elliptical Gaussian vortex positioned at the origin is used to mimic the Kirchhoff vortex

where *A*, *a*, *b*, *n* and *b* is swept to change the aspect ratio. Then, *b* solely (see also **Figure 1b**). *Ro* is reciprocal of *f*. Note again that *Ro* is different among the vortices for the same value of *f* because the velocity and length scales depend on the vortical configuration. The field of **Figure 5** shows examples of initial Gaussian vortex pair with *u* in the

The number of grid points is set to be 2048 × 1024 in the *r* direction (*r*~13,632. The viscosity coefficient and order of viscosity are set to be *p*

In both configurations, the numerical simulations with several *Fr* are performed for cyclone and anticyclone vortex individually, starting from above initial state. Gravity waves are spontaneously radiated from both cyclone and anticyclone vortical flows for large enough *Ro*, while anticyclones rotate in the opposite direction to cyclones. **Figure 6** shows line plots of radiated gravity waves *r* from cyclone and anticyclone for both cases of the corotating vortex pair and Kirchhoff vortex. The line plots in the *Ro* cases. Local maximum appears around *Ro* are remarkably good overlapped between analytical estimates and numerical results, except for the elliptical vortex with for

The intensity of gravity waves for both cases calculated from Eqs. (29) and (30) in the numerical simulation agree well with analytical estimates (not shown). The results indicate that the analytical estimates give the far fields of gravity waves quite accurately and the newly developed numerical model is well verified. Furthermore, cyclone‐anticyclone asymmetry and the local maximum of gravity waves for anticyclone are also confirmed for both cases of the corotating vortex pair and Kirchhoff‐like elliptical vortex. Note that for the elliptical vortex with for

### 3.3. Extended experiments

As extended experiments, the results of the merging of (equal and unequal) vortices are introduced here [24, 25]. As an initial state, a pair of corotating Gaussian vortices expressed by Eq. (42) with different values of *Ro* is reciprocal of *f*. The field of **Figure 7** shows initial Gaussian vortex (cyclone) with

Starting from above initial state, vortices evolve with time. The time evolutions for cyclones with **Figure 8**. Vortices merge in a similar way for both cases of equal vortices and unequal vortices. Initially, vortices corotate with each other (

A series of numerical simulations at different values of *Ro* (*r* values are shown in **Figure 9** for both cyclones and anticyclones with *Ro*.

In order to estimate the intensity of gravity waves quantitatively, a pseudo‐energy flux is derived and calculated as the gravity wave flux [12, 21]. This quantity is conserved when gravity waves propagate into the far field. **Figure 10** shows the maximum values of the gravity wave flux averaged in the *Ro* values (*f* ~ 0.2 for both cases. See Refs. [24] and [25] for details.

## 4. Concluding remarks

### 4.1. Summary points

Far field of inertia‐gravity wave radiated from the corotating point vortex pair and Kirchhoff vortex with nearly circular shape is derived analytically in the

*f*‐plane shallow water system. Cyclone‐anticyclone asymmetry in gravity waves from vortical flows and a local maximum of intensity of gravity waves from anticyclones at an intermediate*f*appear. This is caused by the effect of the Earth's rotation.The derived analytical estimate is well verified for both cases of the corotating vortex pair and Kirchhoff vortex with a small aspect ratio by the numerical simulation with a newly developed spectral method in an unbounded domain.

The numerical experiments extend to the cases of symmetric and asymmetric vortex merger, in which analytical estimate cannot be derived. In both cases, cyclone‐anticyclone asymmetry clearly appears and the local maximum at intermediate

*f*exists only for anticyclones.Within all parameter values and vortical flows used in the present work, there is a cyclone‐anticyclone asymmetry at finite values of

*f*. Gravity waves from anticyclones are larger than those from cyclones and have a local maximum at intermediate*f*. The source originating in the Coriolis acceleration has a key role in cyclone‐anticyclone asymmetry in spontaneous emission. This feature would be robust and ubiquitous in the rotating shallow water system.

### 4.2. Future issues

The derived analytical forms would give useful references for testing the accuracy of the numerical model from a viewpoint of developing new numerical methods.

Cyclone‐anticyclone asymmetry may be related with the change of the dominant balanced state from quasi‐geostrophic one to gradient wind one [29].

There are additional effects which cause the discrepancy between analytical and numerical results, such as nutation of vortex, change of the rotation rate and filaments of the vortex.

More comprehensible understanding of cyclone‐anticyclone asymmetry in spontaneous emission from general complicated vortical flows is needed not only in the rotating shallow water system but also in the continuous stratified system.