2.1 Governing equations

We consider the incompressible Navier–Stokes equations written within the Boussinesq approximation:

(2.1)$$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{u}=0, & \displaystyle\end{eqnarray}$$

(2.2)$$\begin{eqnarray}\displaystyle & \displaystyle \frac{\unicode[STIX]{x2202}\boldsymbol{u}}{\unicode[STIX]{x2202}t}+(\boldsymbol{u}\boldsymbol{\cdot }\unicode[STIX]{x1D735})\boldsymbol{u}=-\unicode[STIX]{x1D735}\left(\frac{p}{\unicode[STIX]{x1D70C}_{0}}\right)+b\boldsymbol{e}_{z}+\unicode[STIX]{x1D708}\unicode[STIX]{x1D6FB}^{2}\boldsymbol{u}, & \displaystyle\end{eqnarray}$$

(2.3)$$\begin{eqnarray}\displaystyle & \displaystyle \frac{\unicode[STIX]{x2202}b}{\unicode[STIX]{x2202}t}+\boldsymbol{u}\boldsymbol{\cdot }\unicode[STIX]{x1D735}b+N^{2}w=\unicode[STIX]{x1D705}\unicode[STIX]{x1D6FB}^{2}b. & \displaystyle\end{eqnarray}$$

Here $\boldsymbol{u}=(u,v,w)$ is the velocity field in Cartesian coordinates $(x,y,z)$, $p$ the pressure, $b=-g\unicode[STIX]{x1D70C}/\unicode[STIX]{x1D70C}_{0}$ the buoyancy, $g$ the gravity, $\boldsymbol{e}_{z}$ the vertical unit vector oriented upwards, $\unicode[STIX]{x1D708}$ the viscosity and $\unicode[STIX]{x1D705}$ the buoyancy diffusivity. The total density field $\unicode[STIX]{x1D70C}_{t}$ has been decomposed as $\unicode[STIX]{x1D70C}_{t}(\boldsymbol{x},t)=\unicode[STIX]{x1D70C}_{0}+\bar{\unicode[STIX]{x1D70C}}(z)+\unicode[STIX]{x1D70C}(\boldsymbol{x},t)$, $\unicode[STIX]{x1D70C}_{0}$ being a constant reference density, $\bar{\unicode[STIX]{x1D70C}}$ a linear mean density profile varying with the vertical coordinate $z$ and $\unicode[STIX]{x1D70C}(\boldsymbol{x},t)$ a perturbation density. The Brunt–Väisälä frequency measuring the ambient stratification $N=\sqrt{-(g/\unicode[STIX]{x1D70C}_{0})\,\text{d}\bar{\unicode[STIX]{x1D70C}}/\text{d}z}$ is assumed constant.

2.2 Initial condition

The initial flow is made of a columnar vortex embedded in a sinusoidal shear flow of the form $\boldsymbol{U}_{S}=U(z)\boldsymbol{e}_{x}$, where $U(z)=\hat{U} _{S}\sin (\hat{k}_{z}z)$, $\boldsymbol{e}_{x}$ is the unit vector in the $x$ direction, $\hat{U} _{S}$ the amplitude and $\hat{k}_{z}$ the vertical wavenumber. The vortex is chosen to have a Lamb–Oseen profile,

(2.4)$$\begin{eqnarray}\boldsymbol{u}_{v}=\frac{\unicode[STIX]{x1D6E4}}{2\unicode[STIX]{x03C0}r}\left[1-\exp \left(-\frac{r^{2}}{a_{0}^{2}}\right)\right]\boldsymbol{e}_{\unicode[STIX]{x1D703}},\end{eqnarray}$$

where $(r,\unicode[STIX]{x1D703},z)$ are cylindrical coordinates such that $x=r\cos (\unicode[STIX]{x1D703})$ and $y=r\sin (\unicode[STIX]{x1D703})$, $\unicode[STIX]{x1D6E4}$ is the vortex circulation, $a_{0}$ its radius and $\boldsymbol{e}_{\unicode[STIX]{x1D703}}$ the unit vector in the azimuthal direction. Without loss of generality, the vortex circulation will be considered positive in the following.

The total initial flow is therefore

(2.5)$$\begin{eqnarray}\boldsymbol{u}(\boldsymbol{x},t=0)=\boldsymbol{U}_{S}+\boldsymbol{u}_{v}.\end{eqnarray}$$

2.3 Non-dimensionalization

Equations (2.1)–(2.3) are non-dimensionalized by the characteristics of the vortex: the length and time units are taken as the vortex radius $a_{0}$ and the turnover time scale $2\unicode[STIX]{x03C0}a_{0}^{2}/\unicode[STIX]{x1D6E4}$. Keeping the same notation for the non-dimensional variables, the dimensionless equations read

(2.6)$$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D735}_{h}\boldsymbol{\cdot }\boldsymbol{u}_{h}+\frac{\unicode[STIX]{x2202}w}{\unicode[STIX]{x2202}z}=0, & \displaystyle\end{eqnarray}$$

(2.7)$$\begin{eqnarray}\displaystyle & \displaystyle \frac{\unicode[STIX]{x2202}\boldsymbol{u}_{h}}{\unicode[STIX]{x2202}t}+\left(\boldsymbol{u}_{h}\boldsymbol{\cdot }\unicode[STIX]{x1D735}_{h}\right)\boldsymbol{u}_{h}+w\frac{\unicode[STIX]{x2202}\boldsymbol{u}_{h}}{\unicode[STIX]{x2202}z}=-\unicode[STIX]{x1D735}_{h}p+\frac{1}{Re}\unicode[STIX]{x1D6FB}^{2}\boldsymbol{u}_{h}, & \displaystyle\end{eqnarray}$$

(2.8)$$\begin{eqnarray}\displaystyle & \displaystyle \frac{\unicode[STIX]{x2202}w}{\unicode[STIX]{x2202}t}+\boldsymbol{u}_{h}\boldsymbol{\cdot }\unicode[STIX]{x1D735}_{h}w+w\frac{\unicode[STIX]{x2202}w}{\unicode[STIX]{x2202}z}=-\frac{\unicode[STIX]{x2202}p}{\unicode[STIX]{x2202}z}+b+\frac{1}{Re}\unicode[STIX]{x1D6FB}^{2}w, & \displaystyle\end{eqnarray}$$

(2.9)$$\begin{eqnarray}\displaystyle & \displaystyle \frac{\unicode[STIX]{x2202}b}{\unicode[STIX]{x2202}t}+\boldsymbol{u}_{h}\boldsymbol{\cdot }\unicode[STIX]{x1D735}_{h}b+w\frac{\unicode[STIX]{x2202}b}{\unicode[STIX]{x2202}z}=-\frac{w}{F_{h}^{2}}+\frac{1}{ReSc}\unicode[STIX]{x1D6FB}^{2}b, & \displaystyle\end{eqnarray}$$

where $\boldsymbol{u}_{h}$ denotes the horizontal velocity. The Reynolds and Froude numbers are defined as

(2.10a,b)$$\begin{eqnarray}Re=\frac{\unicode[STIX]{x1D6E4}}{2\unicode[STIX]{x03C0}\unicode[STIX]{x1D708}},\quad F_{h}=\frac{\unicode[STIX]{x1D6E4}}{2\unicode[STIX]{x03C0}a_{0}^{2}N},\end{eqnarray}$$

and $Sc=\unicode[STIX]{x1D708}/\unicode[STIX]{x1D705}$ is the Schmidt number.

In non-dimensional form, the velocities of the shear flow and the vortex become

(2.11a,b)$$\begin{eqnarray}\boldsymbol{U}_{S}=U_{S}\sin (k_{z}z)\boldsymbol{e}_{x},\quad \boldsymbol{u}_{v}=u_{\unicode[STIX]{x1D703}}(r)\boldsymbol{e}_{\unicode[STIX]{x1D703}}=\frac{1-\exp (-r^{2})}{r}\boldsymbol{e}_{\unicode[STIX]{x1D703}},\end{eqnarray}$$

where $U_{S}=\hat{U} _{S}2\unicode[STIX]{x03C0}a_{0}/\unicode[STIX]{x1D6E4}$ and $k_{z}=\hat{k}_{z}a_{0}$. The corresponding wavelength is $l_{z}=2\unicode[STIX]{x03C0}/k_{z}$.

3.1 Leading-order problem

At leading order, (2.6)–(2.9) reduce to

(3.13)$$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D735}_{h}\boldsymbol{\cdot }\boldsymbol{u}_{h0}=0, & \displaystyle\end{eqnarray}$$

(3.14)$$\begin{eqnarray}\displaystyle & \displaystyle \left(\boldsymbol{u}_{h0}\boldsymbol{\cdot }\unicode[STIX]{x1D735}_{h}\right)\boldsymbol{u}_{h0}=-\unicode[STIX]{x1D735}_{h}p_{0}, & \displaystyle\end{eqnarray}$$

(3.15)$$\begin{eqnarray}\displaystyle & \displaystyle F_{h}^{2}\left[\left(\frac{\unicode[STIX]{x2202}w_{0}}{\unicode[STIX]{x2202}t}\right)_{\tilde{x},{\tilde{y}},\unicode[STIX]{x1D70F}}+\boldsymbol{u}_{h0}\boldsymbol{\cdot }\unicode[STIX]{x1D735}_{h}w_{0}\right]=-\frac{\unicode[STIX]{x2202}p_{0}}{\unicode[STIX]{x2202}\tilde{z}}+b_{0}, & \displaystyle\end{eqnarray}$$

(3.16)$$\begin{eqnarray}\displaystyle & \displaystyle \left(\frac{\unicode[STIX]{x2202}b_{0}}{\unicode[STIX]{x2202}t}\right)_{\tilde{x},{\tilde{y}},\unicode[STIX]{x1D70F}}+\boldsymbol{u}_{h0}\boldsymbol{\cdot }\unicode[STIX]{x1D735}_{h}b_{0}=-w_{0}, & \displaystyle\end{eqnarray}$$

where the operator $\unicode[STIX]{x1D735}_{h}$ is with respect to $\tilde{x}$ and ${\tilde{y}}$, or equivalently $\tilde{r}$ and $\tilde{\unicode[STIX]{x1D703}}$. Note that the advection by the shear flow $U(\tilde{z})\,\unicode[STIX]{x2202}/\unicode[STIX]{x2202}\tilde{x}$ is absent in (3.14)–(3.16) since it cancels with the term coming from the time derivative (see (3.11)).

Equations (3.13)–(3.14) are the steady two-dimensional Euler equations. Consistently with the initial conditions chosen in §2.2, we take an axisymmetric vortex at leading order, namely $\boldsymbol{u}_{h0}=\tilde{r}\unicode[STIX]{x1D6FA}(\tilde{r},\tilde{z},\unicode[STIX]{x1D70F})\boldsymbol{e}_{\tilde{\unicode[STIX]{x1D703}}}$. Even if the angular velocity does not depend on the vertical coordinate $\tilde{z}$ initially, we will see that such dependence appears with time. In order to simplify the notation, $\unicode[STIX]{x1D6FA}$ is not denoted with a subscript $0$ although it is a zeroth-order quantity. Then, equation (3.14) reduces to

(3.17)$$\begin{eqnarray}\frac{\unicode[STIX]{x2202}p_{0}}{\unicode[STIX]{x2202}\tilde{r}}=\tilde{r}\unicode[STIX]{x1D6FA}^{2},\end{eqnarray}$$

which is the balance between the centrifugal force and the radial pressure gradient as if the shear flow were absent. This gives

(3.18)$$\begin{eqnarray}p_{0}=\int ^{\tilde{r}}\unicode[STIX]{x1D709}\unicode[STIX]{x1D6FA}^{2}(\unicode[STIX]{x1D709},\tilde{z},\unicode[STIX]{x1D70F})\,\text{d}\unicode[STIX]{x1D709}.\end{eqnarray}$$

Therefore, the vertical pressure gradient is given by

(3.19)$$\begin{eqnarray}\displaystyle \frac{\unicode[STIX]{x2202}p_{0}}{\unicode[STIX]{x2202}\tilde{z}} & = & \displaystyle \left(\frac{\unicode[STIX]{x2202}p_{0}}{\unicode[STIX]{x2202}\tilde{z}}\right)_{\tilde{r},\tilde{\unicode[STIX]{x1D703}}}+\frac{\unicode[STIX]{x2202}\tilde{r}}{\unicode[STIX]{x2202}\tilde{z}}\frac{\unicode[STIX]{x2202}p_{0}}{\unicode[STIX]{x2202}\tilde{r}}+\frac{\unicode[STIX]{x2202}\tilde{\unicode[STIX]{x1D703}}}{\unicode[STIX]{x2202}\tilde{z}}\frac{\unicode[STIX]{x2202}p_{0}}{\unicode[STIX]{x2202}\tilde{\unicode[STIX]{x1D703}}}\end{eqnarray}$$

(3.20)$$\begin{eqnarray}\displaystyle & = & \displaystyle \int ^{\tilde{r}}\unicode[STIX]{x1D709}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D6FA}^{2}}{\unicode[STIX]{x2202}\tilde{z}}\,\text{d}\unicode[STIX]{x1D709}-\left[\cos (\tilde{\unicode[STIX]{x1D703}})\left(\frac{\text{d}U}{\text{d}\tilde{z}}t+\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D6FF}x}{\unicode[STIX]{x2202}\tilde{z}}\right)+\sin (\tilde{\unicode[STIX]{x1D703}})\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D6FF}y}{\unicode[STIX]{x2202}\tilde{z}}\right]\tilde{r}\unicode[STIX]{x1D6FA}^{2},\end{eqnarray}$$

since (3.5) implies

(3.21)$$\begin{eqnarray}\displaystyle & \displaystyle \frac{\unicode[STIX]{x2202}\tilde{r}}{\unicode[STIX]{x2202}\tilde{z}}=-\cos (\tilde{\unicode[STIX]{x1D703}})\left(\frac{\text{d}U}{\text{d}\tilde{z}}t+\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D6FF}x}{\unicode[STIX]{x2202}\tilde{z}}\right)-\sin (\tilde{\unicode[STIX]{x1D703}})\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D6FF}y}{\unicode[STIX]{x2202}\tilde{z}}, & \displaystyle\end{eqnarray}$$

(3.22)$$\begin{eqnarray}\displaystyle & \displaystyle \frac{\unicode[STIX]{x2202}\tilde{\unicode[STIX]{x1D703}}}{\unicode[STIX]{x2202}\tilde{z}}=\frac{\sin (\tilde{\unicode[STIX]{x1D703}})}{\tilde{r}}\left(\frac{\text{d}U}{\text{d}\tilde{z}}t+\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D6FF}x}{\unicode[STIX]{x2202}\tilde{z}}\right)-\frac{\cos (\tilde{\unicode[STIX]{x1D703}})}{\tilde{r}}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D6FF}y}{\unicode[STIX]{x2202}\tilde{z}}. & \displaystyle\end{eqnarray}$$

Equation (3.20) shows that a vertical pressure gradient appears as soon as the vortex axis is inclined by the ambient shear (second term on the right-hand side) and if the angular velocity $\unicode[STIX]{x1D6FA}$ varies with $\tilde{z}$ (first term). This vertical pressure gradient will force a vertical velocity field and a buoyancy field through (3.15) and (3.16). Since the fluid is strongly stratified, most of the vertical pressure gradient is balanced by buoyancy (i.e. the hydrostatic balance) whereas a vertical velocity is generated mainly to satisfy the buoyancy equation. Using (3.20) and the expression for $\boldsymbol{u}_{h0}$, they can be found as

(3.23)$$\begin{eqnarray}\displaystyle w_{0} & = & \displaystyle {\mathcal{C}}\frac{\text{d}U}{\text{d}\tilde{z}}\left[\cos (\tilde{\unicode[STIX]{x1D703}}){\mathcal{F}}-t\sin (\tilde{\unicode[STIX]{x1D703}})\unicode[STIX]{x1D6FA}-{\mathcal{F}}\cos (\unicode[STIX]{x1D6FA}t-\tilde{\unicode[STIX]{x1D703}})\cos \left(\frac{t}{F_{h}}\right)\right.\nonumber\\ \displaystyle & & \displaystyle -\,\left.2{\mathcal{G}}\sin (\unicode[STIX]{x1D6FA}t-\tilde{\unicode[STIX]{x1D703}})\sin \left(\frac{t}{F_{h}}\right)\right]+\left[-\sin (\tilde{\unicode[STIX]{x1D703}})\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D6FF}x}{\unicode[STIX]{x2202}\tilde{z}}+\cos (\tilde{\unicode[STIX]{x1D703}})\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D6FF}y}{\unicode[STIX]{x2202}\tilde{z}}\right]{\mathcal{C}}\unicode[STIX]{x1D6FA},\qquad\end{eqnarray}$$

(3.24)$$\begin{eqnarray}\displaystyle b_{0} & = & \displaystyle -F_{h}{\mathcal{C}}\frac{\text{d}U}{\text{d}\tilde{z}}\left[2\sin (\tilde{\unicode[STIX]{x1D703}}){\mathcal{G}}+\frac{t}{F_{h}}\cos (\tilde{\unicode[STIX]{x1D703}})+2{\mathcal{G}}\sin (\unicode[STIX]{x1D6FA}t-\tilde{\unicode[STIX]{x1D703}})\cos \left(\frac{t}{F_{h}}\right)\right.\nonumber\\ \displaystyle & & \displaystyle \left.-\,{\mathcal{F}}\cos (\unicode[STIX]{x1D6FA}t-\tilde{\unicode[STIX]{x1D703}})\sin \left(\frac{t}{F_{h}}\right)\right]-\left[\cos (\tilde{\unicode[STIX]{x1D703}})\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D6FF}x}{\unicode[STIX]{x2202}\tilde{z}}+\sin (\tilde{\unicode[STIX]{x1D703}})\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D6FF}y}{\unicode[STIX]{x2202}\tilde{z}}\right]{\mathcal{C}}\nonumber\\ \displaystyle & & \displaystyle +\,\int ^{\tilde{r}}\unicode[STIX]{x1D709}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D6FA}^{2}}{\unicode[STIX]{x2202}\tilde{z}}\,\text{d}\unicode[STIX]{x1D709},\end{eqnarray}$$

where the functions

(3.25a-c)$$\begin{eqnarray}{\mathcal{C}}=\frac{\tilde{r}\unicode[STIX]{x1D6FA}^{2}}{1-F_{h}^{2}\unicode[STIX]{x1D6FA}^{2}},\quad {\mathcal{F}}=\frac{1+F_{h}^{2}\unicode[STIX]{x1D6FA}^{2}}{1-F_{h}^{2}\unicode[STIX]{x1D6FA}^{2}},\quad {\mathcal{G}}=\frac{F_{h}\unicode[STIX]{x1D6FA}}{1-F_{h}^{2}\unicode[STIX]{x1D6FA}^{2}}\end{eqnarray}$$

have no singular point since $\unicode[STIX]{x1D6FA}<1/F_{h}$ for all radii when $F_{h}<1$. We have imposed that the vertical velocity and the buoyancy are null at $t=\unicode[STIX]{x1D70F}=0$ since the vortex is strictly vertical and uniform at the start-up of the evolution.

The vertical velocity and buoyancy equations (3.23) and (3.24) contain four types of terms evolving on three different time scales:

1. (i) The first two terms on the right-hand sides grow linearly with the fast time $t$ in response to the shear flow $U(\tilde{z})$.

2. (ii) Internal waves (third and fourth terms) oscillating at the frequency $1/F_{h}$ (i.e. the Brunt–Väisälä frequency in dimensional units) are generated at the onset of the motion. Strictly speaking, the coefficients in front of these terms are fixed at $\unicode[STIX]{x1D70F}=0$. However, it is also legitimate to let these terms evolve on $\unicode[STIX]{x1D70F}$ like the other terms since the evolution over the slow time scale $\unicode[STIX]{x1D70F}$ is a higher-order effect.

3. (iii) The fifth and sixth terms are due to the slow evolution of the displacement corrections $\unicode[STIX]{x1D6FF}x$ and $\unicode[STIX]{x1D6FF}y$.

4. (iv) A buoyancy perturbation (last term of (3.24)) which is generated by the vertical variation of the angular velocity through the hydrostatic balance.

In the limit $t\ll 1$, $\unicode[STIX]{x1D70F}\ll 1$, (3.23) and (3.24) rescaled by $\unicode[STIX]{x1D700}F_{h}$ and $\unicode[STIX]{x1D700}/F_{h}$ become at leading order

(3.26)$$\begin{eqnarray}\displaystyle & \displaystyle w=\unicode[STIX]{x1D700}F_{h}w_{0}=\frac{1}{2}\cos (\tilde{\unicode[STIX]{x1D703}})\tilde{r}\unicode[STIX]{x1D6FA}^{2}\frac{\text{d}U}{\text{d}z}t^{2}+O(t^{3},\unicode[STIX]{x1D70F}), & \displaystyle\end{eqnarray}$$

(3.27)$$\begin{eqnarray}\displaystyle & \displaystyle b=\unicode[STIX]{x1D700}\frac{b_{0}}{F_{h}}=-\frac{1}{6F_{h}^{2}}\cos (\tilde{\unicode[STIX]{x1D703}})\tilde{r}\unicode[STIX]{x1D6FA}^{2}\frac{\text{d}U}{\text{d}z}t^{3}+O(t^{4},\unicode[STIX]{x1D70F}), & \displaystyle\end{eqnarray}$$

since $\unicode[STIX]{x1D6FF}x$, $\unicode[STIX]{x1D6FF}y$ and the vertical variations of $\unicode[STIX]{x1D6FA}$ are negligible for $\unicode[STIX]{x1D70F}\ll 1$. The vertical velocity and buoyancy are therefore of the form $w\propto t^{2}$ and $b\propto t^{3}$ at leading order in time. By means of an asymptotic expansion for small times, it can be shown that these scaling laws are in fact valid for any finite vertical wavenumber (Bonnici Reference Bonnici2018). Equations (3.26) and (3.27) imply that the ratio of the vertical acceleration to the buoyancy is

(3.28)$$\begin{eqnarray}\left|\frac{1}{b}\frac{\text{D}w}{\text{D}t}\right|=6\left(\frac{F_{h}}{t}\right)^{2}.\end{eqnarray}$$

Therefore, for small times such that $t\ll F_{h}$, the buoyancy is negligible relative to the vertical acceleration, meaning that the dynamics is initially non-hydrostatic. This comes from the generation of internal waves of frequency $1/F_{h}$. This transient non-hydrostatic phase lasts until $t=O(F_{h})$. For $t\gg F_{h}$, the amplitude of the waves in (3.23) and (3.24) will become small relative to the terms that grow linearly with time so that $w\propto F_{h}^{2}t$ and $b\propto t$. The hydrostatic balance will therefore be recovered since $w/b=O(F_{h}^{2})$.

Although the components of the internal waves have been determined explicitly in (3.23) and (3.24), it should be stressed that their asymptotic calculation is actually not uniformly valid when $\unicode[STIX]{x1D700}=k_{z}F_{h}$ is small. It is valid only when $k_{z}$ is small but not if $F_{h}$ is small and $k_{z}$ finite. This limitation comes from the fact that internal waves oscillating at the fast frequency $1/F_{h}$ do not depend on $k_{z}$ and $F_{h}$ only through the self-similar parameter $k_{z}F_{h}$ unlike advective motions in strongly stratified fluids (Billant & Chomaz Reference Billant and Chomaz2001). This can be easily seen by considering the full dispersion relation of internal waves in a medium at rest in Cartesian coordinates, $\unicode[STIX]{x1D714}=k_{h}/(F_{h}\sqrt{k_{h}^{2}+k_{z}^{2}})$, where $k_{h}$ is the horizontal wavenumber, which is assumed to be of order unity. The frequency $\unicode[STIX]{x1D714}$ tends to $1/F_{h}$ if $k_{z}\ll 1$ but not if $k_{z}=O(1)$ and $F_{h}\ll 1$. In the latter case, the internal waves could be computed only numerically. However, we have not attempted such a numerical computation because we will see in Part 2 that the waves actually quickly propagate away from the vortex core when $k_{z}$ is finite. This is due to the fact that their radial group velocity, which is proportional to $k_{z}^{2}/F_{h}$, is large in this case. Thus, their effect in the vortex neighbourhood manifests only initially and is weak so that they can be completely neglected for finite $k_{z}$.

3.2 First-order problem

At first order, it is more convenient to consider the divergence equation and the equation for the vertical vorticity $\unicode[STIX]{x1D701}_{1}=\unicode[STIX]{x2202}v_{1}/\unicode[STIX]{x2202}\tilde{x}-\unicode[STIX]{x2202}u_{1}/\unicode[STIX]{x2202}{\tilde{y}}$:

(3.29)$$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D735}_{h}\boldsymbol{\cdot }\boldsymbol{u}_{h1}=-\left[\frac{\unicode[STIX]{x2202}w_{0}}{\unicode[STIX]{x2202}\tilde{z}}+\frac{\unicode[STIX]{x2202}\tilde{r}}{\unicode[STIX]{x2202}\tilde{z}}\frac{\unicode[STIX]{x2202}w_{0}}{\unicode[STIX]{x2202}\tilde{r}}+\frac{\unicode[STIX]{x2202}\tilde{\unicode[STIX]{x1D703}}}{\unicode[STIX]{x2202}\tilde{z}}\frac{\unicode[STIX]{x2202}w_{0}}{\unicode[STIX]{x2202}\tilde{\unicode[STIX]{x1D703}}}\right], & \displaystyle\end{eqnarray}$$

(3.30)$$\begin{eqnarray}\displaystyle & \displaystyle \frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D701}_{1}}{\unicode[STIX]{x2202}t}+\boldsymbol{u}_{h0}\boldsymbol{\cdot }\unicode[STIX]{x1D735}_{h}\unicode[STIX]{x1D701}_{1}+\boldsymbol{u}_{h1}\boldsymbol{\cdot }\unicode[STIX]{x1D735}_{h}\unicode[STIX]{x1D701}_{0}\hspace{90.00014pt} & \displaystyle \nonumber\\ \displaystyle & \displaystyle =-\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D701}_{0}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D70F}}+\cos (\tilde{\unicode[STIX]{x1D703}})\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D6FF}x}{\unicode[STIX]{x2202}\unicode[STIX]{x1D70F}}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D701}_{0}}{\unicode[STIX]{x2202}\tilde{r}}+\sin (\tilde{\unicode[STIX]{x1D703}})\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D6FF}y}{\unicode[STIX]{x2202}\unicode[STIX]{x1D70F}}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D701}_{0}}{\unicode[STIX]{x2202}\tilde{r}} & \displaystyle \nonumber\\ \displaystyle & \displaystyle \hspace{13.99995pt}+\left[\frac{\text{d}U}{\text{d}\tilde{z}}+\frac{\unicode[STIX]{x2202}\tilde{r}}{\unicode[STIX]{x2202}\tilde{z}}\frac{\unicode[STIX]{x2202}u_{0}}{\unicode[STIX]{x2202}\tilde{r}}+\frac{\unicode[STIX]{x2202}\tilde{\unicode[STIX]{x1D703}}}{\unicode[STIX]{x2202}\tilde{z}}\frac{\unicode[STIX]{x2202}u_{0}}{\unicode[STIX]{x2202}\tilde{\unicode[STIX]{x1D703}}}+\frac{\unicode[STIX]{x2202}u_{0}}{\unicode[STIX]{x2202}\tilde{z}}\right]\frac{\unicode[STIX]{x2202}w_{0}}{\unicode[STIX]{x2202}{\tilde{y}}} & \displaystyle \nonumber\\ \displaystyle & \displaystyle -\left[\frac{\unicode[STIX]{x2202}\tilde{r}}{\unicode[STIX]{x2202}\tilde{z}}\frac{\unicode[STIX]{x2202}v_{0}}{\unicode[STIX]{x2202}\tilde{r}}+\frac{\unicode[STIX]{x2202}\tilde{\unicode[STIX]{x1D703}}}{\unicode[STIX]{x2202}\tilde{z}}\frac{\unicode[STIX]{x2202}v_{0}}{\unicode[STIX]{x2202}\tilde{\unicode[STIX]{x1D703}}}+\frac{\unicode[STIX]{x2202}v_{0}}{\unicode[STIX]{x2202}\tilde{z}}\right]\frac{\unicode[STIX]{x2202}w_{0}}{\unicode[STIX]{x2202}\tilde{x}}\hspace{13.99995pt} & \displaystyle \nonumber\\ \displaystyle & \displaystyle +\,\unicode[STIX]{x1D701}_{0}\left[\frac{\unicode[STIX]{x2202}w_{0}}{\unicode[STIX]{x2202}\tilde{z}}+\frac{\unicode[STIX]{x2202}\tilde{r}}{\unicode[STIX]{x2202}\tilde{z}}\frac{\unicode[STIX]{x2202}w_{0}}{\unicode[STIX]{x2202}\tilde{r}}+\frac{\unicode[STIX]{x2202}\tilde{\unicode[STIX]{x1D703}}}{\unicode[STIX]{x2202}\tilde{z}}\frac{\unicode[STIX]{x2202}w_{0}}{\unicode[STIX]{x2202}\tilde{\unicode[STIX]{x1D703}}}\right]\hspace{16.99998pt} & \displaystyle \nonumber\\ \displaystyle & \displaystyle -\,w_{0}\left[\frac{\unicode[STIX]{x2202}\tilde{r}}{\unicode[STIX]{x2202}\tilde{z}}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D701}_{0}}{\unicode[STIX]{x2202}\tilde{r}}+\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D701}_{0}}{\unicode[STIX]{x2202}\tilde{z}}\right]+\frac{F_{h}^{2}\unicode[STIX]{x1D6FB}^{2}\unicode[STIX]{x1D701}_{0}}{\tilde{Re}_{b}},\hspace{18.99995pt} & \displaystyle\end{eqnarray}$$

where $\unicode[STIX]{x1D701}_{0}=2\unicode[STIX]{x1D6FA}+\tilde{r}\,\unicode[STIX]{x2202}\unicode[STIX]{x1D6FA}/\unicode[STIX]{x2202}\tilde{r}$ is the vertical vorticity. In (3.29) and (3.30), all the time and vertical derivatives are taken at constant $(\tilde{r},\tilde{\unicode[STIX]{x1D703}})$ but this is no longer mentioned to simplify the notation. Such convention will be always used in the following. We remind the reader also that $(u_{0},v_{0},w_{0})$ are the velocity components in Cartesian coordinates.

Equations (3.29) and (3.30) show that the leading-order vertical velocity $w_{0}$ forces a horizontal flow at first order. The forcing terms on the right-hand side of (3.30) correspond to the slow evolution of $\unicode[STIX]{x1D701}_{0}$ and the displacements $\unicode[STIX]{x1D6FF}x$, $\unicode[STIX]{x1D6FF}y$ (first line), the tilting of horizontal vorticity into vertical vorticity (second and third lines), the stretching of vertical vorticity (fourth line), the vertical advection of vertical vorticity and viscous effects (last line). Owing to the expression of the vertical derivative (3.12) (or see (3.19)), the latter term reads

(3.31)$$\begin{eqnarray}\displaystyle F_{h}^{2}\unicode[STIX]{x1D6FB}^{2}\unicode[STIX]{x1D701}_{0} & = & \displaystyle \frac{1}{\tilde{r}}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\tilde{r}}\left(\tilde{r}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D701}_{0}}{\unicode[STIX]{x2202}\tilde{r}}\right)\left[F_{h}^{2}+\frac{\unicode[STIX]{x1D700}^{2}t^{2}}{2}\left(\frac{\text{d}U}{\text{d}\tilde{z}}\right)^{2}\right]+\frac{\unicode[STIX]{x1D700}^{2}t^{2}}{2}\tilde{r}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\tilde{r}}\left(\frac{1}{\tilde{r}}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D701}_{0}}{\unicode[STIX]{x2202}\tilde{r}}\right)\left(\frac{\text{d}U}{\text{d}\tilde{z}}\right)^{2}\cos (2\tilde{\unicode[STIX]{x1D703}})\nonumber\\ \displaystyle & & \displaystyle -\,\unicode[STIX]{x1D700}^{2}t\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D701}_{0}}{\unicode[STIX]{x2202}\tilde{r}}\frac{\text{d}^{2}U}{\text{d}\tilde{z}^{2}}\cos (\tilde{\unicode[STIX]{x1D703}})-2\unicode[STIX]{x1D700}^{2}t\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\tilde{r}}\left[\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D701}_{0}}{\unicode[STIX]{x2202}\tilde{z}}\right]\frac{\text{d}U}{\text{d}\tilde{z}}\cos (\tilde{\unicode[STIX]{x1D703}})+\unicode[STIX]{x1D700}^{2}\frac{\unicode[STIX]{x2202}^{2}\unicode[STIX]{x1D701}_{0}}{\unicode[STIX]{x2202}\tilde{z}^{2}},\end{eqnarray}$$

where we have taken into account only the terms proportional to $t\,\text{d}U/\text{d}\tilde{z}$ in (3.21) and (3.22) and neglected the others. The expression (3.31) suggests that we should neglect all the terms except the first one on the right-hand side since all the others are proportional to $\unicode[STIX]{x1D700}^{2}$. However, some of these terms grow like $t^{2}$ and can thus become of order unity when $\unicode[STIX]{x1D700}t=O(1)$. For this reason, we will take into account in the following the first three terms (first line) which come from the horizontal and vertical dissipative terms and are of order unity or grow like $\tilde{\unicode[STIX]{x1D70F}}^{2}=\unicode[STIX]{x1D700}^{2}t^{2}$, while the last three terms (second line) related to vertical dissipation will be neglected since they are at most of order $\unicode[STIX]{x1D700}$ for $t=O(1/\unicode[STIX]{x1D700})$. Note that, for such large time, the terms proportional to $t\,\text{d}U/\text{d}\tilde{z}$ in (3.21) and (3.22) are much larger than the others, justifying the approximation made to derive (3.31).

We see in (3.31) that the viscous term contains azimuthal modes $m=0$, $1$ and $2$. In fact, the other forcing terms in (3.29) and (3.30) can also be decomposed into these three azimuthal modes. Looking, for example, at the divergence equation (3.29), the term $\unicode[STIX]{x2202}w_{0}/\unicode[STIX]{x2202}\tilde{z}$ will produce terms proportional to $\cos (\tilde{\unicode[STIX]{x1D703}})$ or $\sin (\tilde{\unicode[STIX]{x1D703}})$, while the other two terms will give azimuthal modes of the form $\cos (m\tilde{\unicode[STIX]{x1D703}})$ and $\sin (m\tilde{\unicode[STIX]{x1D703}})$ with $m=0$ or $m=2$. The right-hand side of (3.30) can be decomposed similarly into the azimuthal modes $m=0$, $1$ and $2$. For this reason, (3.29) and (3.30) can be solved by decomposing the horizontal velocity into three potential and rotational components,

(3.32)$$\begin{eqnarray}\boldsymbol{u}_{h1}=\unicode[STIX]{x1D735}_{h}(\unicode[STIX]{x1D6F7}_{10}+\unicode[STIX]{x1D6F7}_{11}+\unicode[STIX]{x1D6F7}_{12})-\unicode[STIX]{x1D735}\times (\unicode[STIX]{x1D713}_{10}+\unicode[STIX]{x1D713}_{11}+\unicode[STIX]{x1D713}_{12})\boldsymbol{e}_{z},\end{eqnarray}$$

corresponding to the azimuthal modes $m=\{0,1,2\}$, respectively. Then, (3.29) and (3.30) each separate into three independent equations

(3.33)$$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D6FB}_{h}^{2}\unicode[STIX]{x1D6F7}_{1m}=\unicode[STIX]{x1D6FF}D_{m}, & \displaystyle\end{eqnarray}$$

(3.34)$$\begin{eqnarray}\displaystyle & \displaystyle \text{L}(\unicode[STIX]{x1D713}_{1m})\equiv \left[\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}t}+\unicode[STIX]{x1D6FA}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\tilde{\unicode[STIX]{x1D703}}}\right]\unicode[STIX]{x1D6FB}_{h}^{2}\unicode[STIX]{x1D713}_{1m}-\frac{1}{\tilde{r}}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D701}_{0}}{\unicode[STIX]{x2202}\tilde{r}}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D713}_{1m}}{\unicode[STIX]{x2202}\tilde{\unicode[STIX]{x1D703}}}=\unicode[STIX]{x1D6FF}Z_{m}-\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D6F7}_{1m}}{\unicode[STIX]{x2202}\tilde{r}}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D701}_{0}}{\unicode[STIX]{x2202}\tilde{r}}, & \displaystyle\end{eqnarray}$$

for $m=\{0,1,2\}$, where $(\unicode[STIX]{x1D6FF}D_{m},\unicode[STIX]{x1D6FF}Z_{m})$ correspond to the terms on the right-hand side of (3.29) and (3.30) with azimuthal mode $m$. We will begin by solving the potential and streamfunction $(\unicode[STIX]{x1D6F7}_{11},\unicode[STIX]{x1D713}_{11})$, which will provide the governing equations for the displacements $(\unicode[STIX]{x1D6FF}x,\unicode[STIX]{x1D6FF}y)$. The potential and streamfunction $(\unicode[STIX]{x1D6F7}_{10},\unicode[STIX]{x1D713}_{10})$ will be determined next in § 3.2.2 and will yield an evolution equation for the angular velocity $\unicode[STIX]{x1D6FA}$, whereas $(\unicode[STIX]{x1D6F7}_{12},\unicode[STIX]{x1D713}_{12})$ are solved in appendix D.

3.2.1 Determination of $\unicode[STIX]{x1D6F7}_{11}$ and $\unicode[STIX]{x1D713}_{11}$

We remind the reader that the angular velocity of the vortex is assumed to vary over $\unicode[STIX]{x1D70F}$ and $\tilde{z}$. However, these dependences are unknown at this stage since they will be determined only in § 3.2.2. Equation (3.33) for the potential $\unicode[STIX]{x1D6F7}_{11}$ is

(3.35)$$\begin{eqnarray}\unicode[STIX]{x1D6FB}_{h}^{2}\unicode[STIX]{x1D6F7}_{11}=-\frac{\unicode[STIX]{x2202}w_{0}}{\unicode[STIX]{x2202}\tilde{z}}.\end{eqnarray}$$

By imposing that $\unicode[STIX]{x1D6F7}_{11}$ is not singular at $\tilde{r}=0$ and vanishes as $\tilde{r}$ tends to infinity, the solution can be found by the method of variation of parameters as

(3.36)$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6F7}_{11} & = & \displaystyle \frac{1}{2}\left[\sin (\tilde{\unicode[STIX]{x1D703}})\frac{\unicode[STIX]{x2202}^{2}\unicode[STIX]{x1D6FF}x}{\unicode[STIX]{x2202}\tilde{z}^{2}}-\cos (\tilde{\unicode[STIX]{x1D703}})\frac{\unicode[STIX]{x2202}^{2}\unicode[STIX]{x1D6FF}y}{\unicode[STIX]{x2202}\tilde{z}^{2}}\right]\unicode[STIX]{x1D6F7}_{s}+\frac{1}{2}t\sin (\tilde{\unicode[STIX]{x1D703}})\unicode[STIX]{x1D6F7}_{p\unicode[STIX]{x1D6FA}}\frac{\text{d}U}{\text{d}\tilde{z}}\nonumber\\ \displaystyle & & \displaystyle +\,\frac{1}{2}\left[-\cos (\tilde{\unicode[STIX]{x1D703}})\unicode[STIX]{x1D6F7}_{pc}+t\sin (\tilde{\unicode[STIX]{x1D703}})\unicode[STIX]{x1D6F7}_{ps}\right]\frac{\text{d}^{2}U}{\text{d}\tilde{z}^{2}}\nonumber\\ \displaystyle & & \displaystyle -\,\left[\unicode[STIX]{x1D6F7}_{wc}\cos (\tilde{\unicode[STIX]{x1D703}})+\unicode[STIX]{x1D6F7}_{ws}\sin (\tilde{\unicode[STIX]{x1D703}})\right]\frac{\text{d}^{2}U}{\text{d}\tilde{z}^{2}},\end{eqnarray}$$

where $\unicode[STIX]{x1D6F7}_{s}$, $\unicode[STIX]{x1D6F7}_{p\unicode[STIX]{x1D6FA}}$, $\unicode[STIX]{x1D6F7}_{ps}$, $\unicode[STIX]{x1D6F7}_{pc}$, $\unicode[STIX]{x1D6F7}_{wc}$ and $\unicode[STIX]{x1D6F7}_{ws}$ are defined in appendix B. The terms associated with these potentials correspond, respectively, to the effects of the displacements ($\unicode[STIX]{x1D6F7}_{s}$), the vertical variations of the angular velocity ($\unicode[STIX]{x1D6F7}_{p\unicode[STIX]{x1D6FA}}$), the shear flow ($\unicode[STIX]{x1D6F7}_{ps}$, $\unicode[STIX]{x1D6F7}_{pc}$) and the internal waves ($\unicode[STIX]{x1D6F7}_{wc}$, $\unicode[STIX]{x1D6F7}_{ws}$). The vertical derivative of the angular velocity has been taken into account only in the leading-order term of the vertical velocity when $t$ is large in order to compute the second term in the first line of (3.36). Indeed, since the vertical variations of $\unicode[STIX]{x1D6FA}$ are significant only when $t$ is large, at that time the vertical velocity (3.23) can be approximated by $w_{0}\simeq -t\sin (\tilde{\unicode[STIX]{x1D703}})\unicode[STIX]{x1D6FA}{\mathcal{C}}\,\text{d}U/\text{d}\tilde{z}$.

Equation (3.34) for the streamfunction $\unicode[STIX]{x1D713}_{11}$ is

(3.37)$$\begin{eqnarray}\displaystyle \text{L}(\unicode[STIX]{x1D713}_{11}) & = & \displaystyle \left[\cos (\tilde{\unicode[STIX]{x1D703}})\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D6FF}x}{\unicode[STIX]{x2202}\unicode[STIX]{x1D70F}}+\sin (\tilde{\unicode[STIX]{x1D703}})\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D6FF}y}{\unicode[STIX]{x2202}\unicode[STIX]{x1D70F}}-\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D6F7}_{11}}{\unicode[STIX]{x2202}\tilde{r}}\right]\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D701}_{0}}{\unicode[STIX]{x2202}\tilde{r}}\nonumber\\ \displaystyle & & \displaystyle +\,\unicode[STIX]{x1D701}_{0}\frac{\unicode[STIX]{x2202}w_{0}}{\unicode[STIX]{x2202}\tilde{z}}-w_{0}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D701}_{0}}{\unicode[STIX]{x2202}\tilde{z}}-\tilde{r}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D6FA}}{\unicode[STIX]{x2202}\tilde{z}}\frac{\unicode[STIX]{x2202}w_{0}}{\unicode[STIX]{x2202}\tilde{r}}.\end{eqnarray}$$

The first two terms on the left-hand side of (3.37) come from the slow evolution of $\unicode[STIX]{x1D6FF}x$ and $\unicode[STIX]{x1D6FF}y$. The third term is the advection of $\unicode[STIX]{x1D701}_{0}$ by the first-order potential flow. The fourth term is the stretching of $\unicode[STIX]{x1D701}_{0}$. The fifth term comes from the vertical advection of $\unicode[STIX]{x1D701}_{0}$. The last term corresponds to the tilting of horizontal vorticity due to the vertical variation of the angular velocity. The solution, which is non-singular at $\tilde{r}=0$, can also be found by the method of variation of parameters as

(3.38)$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D713}_{11} & = & \displaystyle \left[\cos (\tilde{\unicode[STIX]{x1D703}})\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D6FF}y}{\unicode[STIX]{x2202}\unicode[STIX]{x1D70F}}-\sin (\tilde{\unicode[STIX]{x1D703}})\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D6FF}x}{\unicode[STIX]{x2202}\unicode[STIX]{x1D70F}}\right]\left(\tilde{r}+\unicode[STIX]{x1D707}\tilde{r}\unicode[STIX]{x1D6FA}\right)\nonumber\\ \displaystyle & & \displaystyle +\,\left[\cos (\tilde{\unicode[STIX]{x1D703}})\frac{\unicode[STIX]{x2202}^{2}\unicode[STIX]{x1D6FF}x}{\unicode[STIX]{x2202}\tilde{z}^{2}}+\sin (\tilde{\unicode[STIX]{x1D703}})\frac{\unicode[STIX]{x2202}^{2}\unicode[STIX]{x1D6FF}y}{\unicode[STIX]{x2202}\tilde{z}^{2}}\right]\left(\unicode[STIX]{x1D713}_{s}+\unicode[STIX]{x1D6FC}\tilde{r}\unicode[STIX]{x1D6FA}\right)\nonumber\\ \displaystyle & & \displaystyle +\,t\cos (\tilde{\unicode[STIX]{x1D703}})(\unicode[STIX]{x1D713}_{p\unicode[STIX]{x1D6FA}}+\unicode[STIX]{x1D706}\tilde{r}\unicode[STIX]{x1D6FA})\frac{\text{d}U}{\text{d}\tilde{z}}\nonumber\\ \displaystyle & & \displaystyle -\,\left[\vphantom{\frac{\unicode[STIX]{x2202}{\mathcal{C}}_{w}}{\unicode[STIX]{x2202}t}}\sin (\tilde{\unicode[STIX]{x1D703}})\left(\unicode[STIX]{x1D706}_{0}\tilde{r}\unicode[STIX]{x1D6FA}+\unicode[STIX]{x1D706}_{1}\tilde{r}+\unicode[STIX]{x1D713}_{ps}\right)+t\cos (\tilde{\unicode[STIX]{x1D703}})\left(\unicode[STIX]{x1D706}_{1}\tilde{r}\unicode[STIX]{x1D6FA}+\unicode[STIX]{x1D713}_{pc}\right)\right.\nonumber\\ \displaystyle & & \displaystyle -\,\cos (\tilde{\unicode[STIX]{x1D703}})\left(\unicode[STIX]{x1D713}_{wc}+\tilde{r}\unicode[STIX]{x1D6FA}{\mathcal{C}}_{w}-\tilde{r}\frac{\unicode[STIX]{x2202}{\mathcal{S}}_{w}}{\unicode[STIX]{x2202}t}\right)\nonumber\\ \displaystyle & & \displaystyle \left.-\,\sin (\tilde{\unicode[STIX]{x1D703}})\left(\unicode[STIX]{x1D713}_{ws}+\tilde{r}\unicode[STIX]{x1D6FA}{\mathcal{S}}_{w}+\tilde{r}\frac{\unicode[STIX]{x2202}{\mathcal{C}}_{w}}{\unicode[STIX]{x2202}t}\right)\right]\frac{\text{d}^{2}U}{\text{d}\tilde{z}^{2}},\end{eqnarray}$$

where $\unicode[STIX]{x1D713}_{s}$, $\unicode[STIX]{x1D713}_{p\unicode[STIX]{x1D6FA}}$, $\unicode[STIX]{x1D713}_{ps}$ and $\unicode[STIX]{x1D713}_{pc}$ are given in appendix C, while $\unicode[STIX]{x1D713}_{wc}$ and $\unicode[STIX]{x1D713}_{ws}$ will be defined later. The subscripts on these streamfunctions denote the same physical effects as for the potentials in (3.36). The parameters $\unicode[STIX]{x1D707}$, $\unicode[STIX]{x1D6FC}$, $\unicode[STIX]{x1D706}$, $\unicode[STIX]{x1D706}_{0}$ and $\unicode[STIX]{x1D706}_{1}$ are arbitrary functions of $\tilde{z}$ and $\unicode[STIX]{x1D70F}$, while ${\mathcal{C}}_{w}$ and ${\mathcal{S}}_{w}$ are arbitrary functions of $\tilde{z}$, $\unicode[STIX]{x1D70F}$ and $t$. These functions are free because

(3.39)$$\begin{eqnarray}\unicode[STIX]{x1D713}_{h}=\tilde{r}\unicode[STIX]{x1D6FA}[\cos (\tilde{\unicode[STIX]{x1D703}})f(t)+\sin (\tilde{\unicode[STIX]{x1D703}})g(t)]+\tilde{r}[\sin (\tilde{\unicode[STIX]{x1D703}})f^{\prime }(t)-\cos (\tilde{\unicode[STIX]{x1D703}})g^{\prime }(t)]\end{eqnarray}$$

is a homogeneous solution of (3.37) for any $f$ and $g$ depending on $t$. These solutions derive from the translational invariance (Billant Reference Billant2010). In order to fix these parameters, we need to use a normalization condition. Here, we have chosen to impose that the mean displacement associated with the first-order vorticity $\unicode[STIX]{x1D6FB}_{h}^{2}\unicode[STIX]{x1D713}_{11}$ is zero, i.e.

(3.40a,b)$$\begin{eqnarray}\int _{0}^{2\unicode[STIX]{x03C0}}\int _{0}^{+\infty }x\unicode[STIX]{x1D6FB}_{h}^{2}\unicode[STIX]{x1D713}_{11}\tilde{r}\,\text{d}\tilde{r}\,\text{d}\tilde{\unicode[STIX]{x1D703}}=0,\quad \int _{0}^{2\unicode[STIX]{x03C0}}\int _{0}^{+\infty }y\unicode[STIX]{x1D6FB}_{h}^{2}\unicode[STIX]{x1D713}_{11}\tilde{r}\,\text{d}\tilde{r}\,\text{d}\tilde{\unicode[STIX]{x1D703}}=0.\end{eqnarray}$$

In this way, only the displacements $U(\tilde{z})t+\unicode[STIX]{x1D6FF}x$ and $\unicode[STIX]{x1D6FF}y$ of the leading-order vorticity $\unicode[STIX]{x1D701}_{0}$ will contribute to the mean displacement of the vortex. This will ease the comparison with the numerical simulations performed in Part 2, where the vortex displacements will be computed from vertical vorticity centroids. Hence, to enforce (3.40), we impose that

(3.41)$$\begin{eqnarray}\displaystyle \langle \unicode[STIX]{x1D6FB}_{h}^{2}\unicode[STIX]{x1D713}_{11},\tilde{r}\rangle \equiv \int _{0}^{+\infty }\unicode[STIX]{x1D6FB}_{h}^{2}\unicode[STIX]{x1D713}_{11}\tilde{r}^{2}\,\text{d}\tilde{r}=0. & & \displaystyle\end{eqnarray}$$

In practice, we shall impose that each individual effect in (3.38) satisfies this condition. This leads to

(3.42a-d)$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D707}=0,\quad \unicode[STIX]{x1D706}_{1}=-\unicode[STIX]{x1D6FC}={\textstyle \frac{1}{2}}\langle \unicode[STIX]{x1D6FB}_{\tilde{r}}^{2}\unicode[STIX]{x1D713}_{pc},\tilde{r}\rangle ,\quad \unicode[STIX]{x1D706}={\textstyle \frac{1}{2}}\langle \unicode[STIX]{x1D6FB}_{\tilde{r}}^{2}\unicode[STIX]{x1D713}_{p\unicode[STIX]{x1D6FA}},\tilde{r}\rangle ,\quad \unicode[STIX]{x1D706}_{0}={\textstyle \frac{1}{2}}\langle \unicode[STIX]{x1D6FB}_{\tilde{r}}^{2}\unicode[STIX]{x1D713}_{ps},\tilde{r}\rangle ,\qquad \qquad & & \displaystyle\end{eqnarray}$$

since $\langle \unicode[STIX]{x1D6FB}_{\tilde{r}}^{2}(\tilde{r}\unicode[STIX]{x1D6FA}),\tilde{r}\rangle =-2$, where the operator $\unicode[STIX]{x1D6FB}_{\tilde{r}}^{2}$ is given by

(3.43)$$\begin{eqnarray}\unicode[STIX]{x1D6FB}_{\tilde{r}}^{2}\unicode[STIX]{x1D6EC}=\frac{\unicode[STIX]{x2202}^{2}\unicode[STIX]{x1D6EC}}{\unicode[STIX]{x2202}\tilde{r}^{2}}+\frac{1}{\tilde{r}}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D6EC}}{\unicode[STIX]{x2202}\tilde{r}}-\frac{\unicode[STIX]{x1D6EC}}{\tilde{r}^{2}},\end{eqnarray}$$

for any function $\unicode[STIX]{x1D6EC}$. Finally, the two streamfunctions $\unicode[STIX]{x1D713}_{wc}$ and $\unicode[STIX]{x1D713}_{ws}$ associated with internal waves satisfy

(3.44)$$\begin{eqnarray}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D6FB}_{\tilde{r}}^{2}\unicode[STIX]{x1D713}_{wc}}{\unicode[STIX]{x2202}t}+\unicode[STIX]{x1D6FA}\unicode[STIX]{x1D6FB}_{\tilde{r}}^{2}\unicode[STIX]{x1D713}_{ws}-\frac{1}{\tilde{r}}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D701}_{0}}{\unicode[STIX]{x2202}\tilde{r}}\unicode[STIX]{x1D713}_{ws}=-\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D701}_{0}}{\unicode[STIX]{x2202}\tilde{r}}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D6F7}_{wc}}{\unicode[STIX]{x2202}\tilde{r}}-\unicode[STIX]{x1D701}_{0}\unicode[STIX]{x1D6FB}_{\tilde{r}}^{2}\unicode[STIX]{x1D6F7}_{wc},\end{eqnarray}$$

(3.45)$$\begin{eqnarray}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D6FB}_{\tilde{r}}^{2}\unicode[STIX]{x1D713}_{ws}}{\unicode[STIX]{x2202}t}-\unicode[STIX]{x1D6FA}\unicode[STIX]{x1D6FB}_{\tilde{r}}^{2}\unicode[STIX]{x1D713}_{wc}+\frac{1}{\tilde{r}}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D701}_{0}}{\unicode[STIX]{x2202}\tilde{r}}\unicode[STIX]{x1D713}_{wc}=-\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D701}_{0}}{\unicode[STIX]{x2202}\tilde{r}}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D6F7}_{ws}}{\unicode[STIX]{x2202}\tilde{r}}-\unicode[STIX]{x1D701}_{0}\unicode[STIX]{x1D6FB}_{\tilde{r}}^{2}\unicode[STIX]{x1D6F7}_{ws},\end{eqnarray}$$

where the potentials $\unicode[STIX]{x1D6F7}_{wc}$ and $\unicode[STIX]{x1D6F7}_{ws}$ are given by (B 2) and (B 3) in appendix B. Because the time and radial dependences in $\unicode[STIX]{x1D6F7}_{wc}$ and $\unicode[STIX]{x1D6F7}_{ws}$ are not separated, it is not possible to solve (3.44) and (3.45) analytically. We have therefore implemented a numerical resolution of (3.44) and (3.45). The time advancement is performed by a forward Euler scheme, spatial derivatives are computed by centred finite differences, and the Laplacian operator is inverted by means of the tridiagonal algorithm. The evolutions of $\unicode[STIX]{x1D6FA}$ and $\unicode[STIX]{x1D701}_{0}$ over the slow time are taken into account through the numerical integration of equation (4.2), which will be derived later. The imposed boundary conditions are that $\unicode[STIX]{x1D713}_{ws}=\unicode[STIX]{x1D713}_{wc}=0$ at $\tilde{r}=0$ and that $\unicode[STIX]{x1D713}_{ws}$ and $\unicode[STIX]{x1D713}_{wc}$ vanish as $\tilde{r}\rightarrow +\infty$. Since the total streamfunction $\unicode[STIX]{x1D713}_{11}$ defined in (3.38) should be zero at $t=\unicode[STIX]{x1D70F}=0$, the initial condition should be

(3.46a,b)$$\begin{eqnarray}\unicode[STIX]{x1D713}_{wc}=0,\quad \unicode[STIX]{x1D713}_{ws}=-[\unicode[STIX]{x1D713}_{ps}+\unicode[STIX]{x1D706}_{0}\tilde{r}\unicode[STIX]{x1D6FA}+\unicode[STIX]{x1D706}_{1}\tilde{r}-\unicode[STIX]{x1D717}(\tilde{z},\unicode[STIX]{x1D70F}=0)\tilde{r}],\end{eqnarray}$$

since $\unicode[STIX]{x1D6FF}x=\unicode[STIX]{x1D6FF}y=\unicode[STIX]{x2202}\unicode[STIX]{x1D6FF}x/\unicode[STIX]{x2202}\unicode[STIX]{x1D70F}=\unicode[STIX]{x2202}\unicode[STIX]{x1D6FF}y/\unicode[STIX]{x2202}\unicode[STIX]{x1D70F}=0$ at $\unicode[STIX]{x1D70F}=0$. The parameter $\unicode[STIX]{x1D717}(\tilde{z},\unicode[STIX]{x1D70F}=0)$ is set such that $\unicode[STIX]{x1D713}_{ws}\rightarrow 0$ as $\tilde{r}\rightarrow +\infty$ initially, as required to integrate (3.44) and (3.45) numerically. From the expression (C 3) of $\unicode[STIX]{x1D713}_{ps}$, we have

(3.47)$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D717}(\tilde{z},\unicode[STIX]{x1D70F}) & = & \displaystyle \unicode[STIX]{x1D706}_{1}(\tilde{z},\unicode[STIX]{x1D70F})-\int _{0}^{+\infty }{\mathcal{C}}\left(\unicode[STIX]{x1D709}^{2}\unicode[STIX]{x1D6FA}^{2}F_{h}{\mathcal{G}}-\frac{1}{2}\right)\,\text{d}\unicode[STIX]{x1D709}\nonumber\\ \displaystyle & & \displaystyle -\,\int _{0}^{+\infty }\frac{\unicode[STIX]{x1D701}_{0}}{\unicode[STIX]{x1D702}^{3}\unicode[STIX]{x1D6FA}^{3}}\int _{0}^{\unicode[STIX]{x1D702}}\unicode[STIX]{x1D709}^{2}\unicode[STIX]{x1D6FA}^{2}{\mathcal{C}}\,\text{d}\unicode[STIX]{x1D709}\,\text{d}\unicode[STIX]{x1D702},\end{eqnarray}$$

where ${\mathcal{C}}$ and ${\mathcal{G}}$ are defined in (3.25). There is no loss of generality in imposing that $\unicode[STIX]{x1D713}_{ws}$ vanishes as $\tilde{r}\rightarrow +\infty$ initially because $\unicode[STIX]{x1D713}_{wh}=\unicode[STIX]{x1D717}t\tilde{r}\unicode[STIX]{x1D6FA}\cos (\tilde{\unicode[STIX]{x1D703}})+\unicode[STIX]{x1D717}\tilde{r}\sin (\tilde{\unicode[STIX]{x1D703}})$ is a homogeneous solution of (3.37). In other words, introducing the term $\unicode[STIX]{x1D717}\tilde{r}$ in $\unicode[STIX]{x1D713}_{ws}$ is equivalent to substituting ${\mathcal{C}}_{w}$ by ${\mathcal{C}}_{w}+\unicode[STIX]{x1D717}t$ in (3.38). After the numerical integration, we enforce that the mean displacement associated to the internal waves is zero, yielding

(3.48a,b)$$\begin{eqnarray}{\mathcal{C}}_{w}(t,\tilde{z},\unicode[STIX]{x1D70F})={\textstyle \frac{1}{2}}\langle \unicode[STIX]{x1D6FB}_{\tilde{r}}^{2}\unicode[STIX]{x1D713}_{wc},\tilde{r}\rangle ,\quad {\mathcal{S}}_{w}(t,\tilde{z},\unicode[STIX]{x1D70F})={\textstyle \frac{1}{2}}\langle \unicode[STIX]{x1D6FB}_{\tilde{r}}^{2}\unicode[STIX]{x1D713}_{ws},\tilde{r}\rangle .\end{eqnarray}$$

Figure 2(a,b) shows the streamfunctions $\unicode[STIX]{x1D713}_{wc}$ and $\unicode[STIX]{x1D713}_{ws}$ plotted against $\tilde{r}$ at different times, for $F_{h}=0.1$ at $z=l_{z}/4$. In the next section, we will show that the angular velocity $\unicode[STIX]{x1D6FA}$ does not evolve with $\unicode[STIX]{x1D70F}$ in this plane. Hence, (3.44) and (3.45) have been integrated by imposing the initial angular velocity profile given by (2.11) for all time. We see that, after an initial transient phase, the streamfunctions globally decay with time in the vortex core.

Figure 2.

Streamfunctions (a) $\unicode[STIX]{x1D713}_{wc}$ and (b) $\unicode[STIX]{x1D713}_{ws}$ plotted against $\tilde{r}$ at $t=0$ (grey solid line), $t=4$ (black dash-dotted line), $t=8$ (black dashed line) and $t=16$ (black solid line), at $z=l_{z}/4$, where $\unicode[STIX]{x1D6FA}=[1-\exp (-\tilde{r}^{2})]/\tilde{r}^{2}$ regardless of $\unicode[STIX]{x1D70F}$. The Froude number is $F_{h}=0.1$. (c) Streamfunctions $\bar{\unicode[STIX]{x1D713}}_{2b}/(\unicode[STIX]{x1D6FD}_{2}\bar{r}^{2})$ (solid line), $\bar{\unicode[STIX]{x1D713}}_{2s}/(\unicode[STIX]{x1D70E}_{2}\bar{r}^{2})$ (dashed line), $\bar{\unicode[STIX]{x1D713}}_{2c}/(\unicode[STIX]{x1D712}_{2}\bar{r}^{2})$ (dash-dotted line) and $\bar{\unicode[STIX]{x1D713}}_{2s}^{\unicode[STIX]{x1D708}}/(\unicode[STIX]{x1D70E}_{2}^{\unicode[STIX]{x1D708}}\bar{r}^{2})$ (dotted line) computed in appendix D.

We have now fully determined the streamfunction at first order $\unicode[STIX]{x1D713}_{11}$. It satisfies the boundary condition at $\tilde{r}=0$. However, it can be shown that it behaves for large $\tilde{r}$ as

(3.49)$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D713}_{11} & {\sim} & \displaystyle \left[\cos (\tilde{\unicode[STIX]{x1D703}})\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D6FF}y}{\unicode[STIX]{x2202}\unicode[STIX]{x1D70F}}-\sin (\tilde{\unicode[STIX]{x1D703}})\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D6FF}x}{\unicode[STIX]{x2202}\unicode[STIX]{x1D70F}}\right]\tilde{r}\nonumber\\ \displaystyle & & \displaystyle +\,\frac{{\mathcal{A}}}{2}\left[\cos (\tilde{\unicode[STIX]{x1D703}})\frac{\unicode[STIX]{x2202}^{2}\unicode[STIX]{x1D6FF}x}{\unicode[STIX]{x2202}\tilde{z}^{2}}+\sin (\tilde{\unicode[STIX]{x1D703}})\frac{\unicode[STIX]{x2202}^{2}\unicode[STIX]{x1D6FF}y}{\unicode[STIX]{x2202}\tilde{z}^{2}}\right]\tilde{r}+\cos (\tilde{\unicode[STIX]{x1D703}})\frac{{\mathcal{B}}t}{2}\frac{\text{d}U}{\text{d}\tilde{z}}\tilde{r}\nonumber\\ \displaystyle & & \displaystyle +\,\left[-\sin (\tilde{\unicode[STIX]{x1D703}})\unicode[STIX]{x1D717}+\cos (\tilde{\unicode[STIX]{x1D703}})\frac{{\mathcal{A}}t}{2}+\cos (\tilde{\unicode[STIX]{x1D703}})\frac{\unicode[STIX]{x2202}{\mathcal{S}}_{w}}{\unicode[STIX]{x2202}t}-\sin (\tilde{\unicode[STIX]{x1D703}})\frac{\unicode[STIX]{x2202}{\mathcal{C}}_{w}}{\unicode[STIX]{x2202}t}\right]\frac{\text{d}^{2}U}{\text{d}\tilde{z}^{2}}\tilde{r},\end{eqnarray}$$

where ${\mathcal{A}}$ and ${\mathcal{B}}$ are given by

(3.50)$$\begin{eqnarray}\displaystyle & \displaystyle {\mathcal{A}}(\tilde{z},\unicode[STIX]{x1D70F})=\int _{0}^{+\infty }\unicode[STIX]{x1D709}^{2}\unicode[STIX]{x1D6FA}^{2}{\mathcal{C}}\,\text{d}\unicode[STIX]{x1D709}, & \displaystyle\end{eqnarray}$$

(3.51)$$\begin{eqnarray}\displaystyle & \displaystyle {\mathcal{B}}(\tilde{z},\unicode[STIX]{x1D70F})=\int _{0}^{+\infty }\unicode[STIX]{x1D709}^{2}\unicode[STIX]{x1D6FA}{\mathcal{C}}\left(3+{\mathcal{F}}\right)\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D6FA}}{\unicode[STIX]{x2202}\tilde{z}}\,\text{d}\unicode[STIX]{x1D709}\equiv \frac{\unicode[STIX]{x2202}{\mathcal{A}}}{\unicode[STIX]{x2202}\tilde{z}}, & \displaystyle\end{eqnarray}$$

where ${\mathcal{C}}$ and ${\mathcal{F}}$ are defined in (3.25). Note that we have used the fact that $\unicode[STIX]{x1D713}_{wc}$ and $\unicode[STIX]{x1D713}_{ws}$ tend to zero as $\tilde{r}\rightarrow +\infty$.

In order to enforce the boundary condition $\unicode[STIX]{x1D713}_{11}\rightarrow 0$ as $\tilde{r}\rightarrow +\infty$, we need to impose

(3.52)$$\begin{eqnarray}\displaystyle & \displaystyle \frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D6FF}x}{\unicode[STIX]{x2202}\unicode[STIX]{x1D70F}}=\left[\frac{{\mathcal{A}}(\tilde{z},\unicode[STIX]{x1D70F})}{2}\frac{\unicode[STIX]{x2202}^{2}\unicode[STIX]{x1D6FF}y}{\unicode[STIX]{x2202}\tilde{z}^{2}}-\left(\unicode[STIX]{x1D717}(\tilde{z},\unicode[STIX]{x1D70F})+\frac{\unicode[STIX]{x2202}{\mathcal{C}}_{w}}{\unicode[STIX]{x2202}t}(t,\tilde{z},\unicode[STIX]{x1D70F})\right)\frac{\text{d}^{2}U}{\text{d}\tilde{z}^{2}}\right], & \displaystyle\end{eqnarray}$$

(3.53)$$\begin{eqnarray}\displaystyle & \displaystyle \frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D6FF}y}{\unicode[STIX]{x2202}\unicode[STIX]{x1D70F}}=-\left[\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\tilde{z}}\left(\frac{{\mathcal{A}}(\tilde{z},\unicode[STIX]{x1D70F})}{2}t\frac{\text{d}U}{\text{d}\tilde{z}}\right)+\frac{{\mathcal{A}}(\tilde{z},\unicode[STIX]{x1D70F})}{2}\frac{\unicode[STIX]{x2202}^{2}\unicode[STIX]{x1D6FF}x}{\unicode[STIX]{x2202}\tilde{z}^{2}}+\frac{\unicode[STIX]{x2202}{\mathcal{S}}_{w}}{\unicode[STIX]{x2202}t}(t,\tilde{z},\unicode[STIX]{x1D70F})\frac{\text{d}^{2}U}{\text{d}\tilde{z}^{2}}\right]. & \displaystyle\end{eqnarray}$$

When $U=0$, these equations are identical to those found by Billant (Reference Billant2010) for the self-induction of an isolated vortex in stratified non-rotating fluids.

Figure 3(a) shows ${\mathcal{A}}$ and $\unicode[STIX]{x1D717}$ as functions of $F_{h}$ at $z=l_{z}/4$, where the angular velocity remains the same, $\unicode[STIX]{x1D6FA}=[1-\exp (-\tilde{r}^{2})]/\tilde{r}^{2}$, independent of $\unicode[STIX]{x1D70F}$ (see § 4.1). They are almost independent of $F_{h}$ for $F_{h}\lesssim 0.5$. The wave forcing terms ${\mathcal{C}}_{w}$ and ${\mathcal{S}}_{w}$ are also shown in figure 3(b,c) for two Froude numbers $F_{h}=0.1$ and $F_{h}=0.5$, at the same location. In appendix E, it is shown that they behave for small time as

(3.54a,b)$$\begin{eqnarray}{\mathcal{C}}_{w}=-\unicode[STIX]{x1D717}t+O(t^{3}),\quad {\mathcal{S}}_{w}=-\frac{{\mathcal{A}}}{4}t^{2}+O(t^{4}).\end{eqnarray}$$

Therefore, they initially compensate the effect of the shear flow in (3.52) and (3.53). Then, they go through oscillations as seen in figure 3(b,c). These oscillations exhibit two typical periods: the non-dimensional period of rotation in the vortex core, $T\sim 2\unicode[STIX]{x03C0}$, and the buoyancy period, $T\sim 2\unicode[STIX]{x03C0}F_{h}$. For $F_{h}=0.5$ (figure 3c), these two periods are comparable so that ${\mathcal{C}}_{w}$ and ${\mathcal{S}}_{w}$ exhibit irregular oscillations, while for $F_{h}=0.1$ (figure 3b), fast oscillations superimposed on slow oscillations can be clearly distinguished. For large times, ${\mathcal{C}}_{w}$ and ${\mathcal{S}}_{w}$ tend to constant values because the waves are increasingly sheared by the vortex rotation through the terms $\cos (\unicode[STIX]{x1D6FA}t-\tilde{\unicode[STIX]{x1D703}})$ and $\sin (\unicode[STIX]{x1D6FA}t-\tilde{\unicode[STIX]{x1D703}})$ as seen in (3.23). As a result, their contributions $\unicode[STIX]{x2202}{\mathcal{C}}_{w}/\unicode[STIX]{x2202}t$ and $\unicode[STIX]{x2202}{\mathcal{S}}_{w}/\unicode[STIX]{x2202}t$ to the vortex displacements vanish as $t\rightarrow +\infty$.

Figure 3.

(a) Variables ${\mathcal{A}}$ (solid line) and $\unicode[STIX]{x1D717}$ (dashed line) plotted against $F_{h}$, at the vertical level $z=l_{z}/4$, where $\unicode[STIX]{x1D6FA}=[1-\exp (-\tilde{r}^{2})]/\tilde{r}^{2}$ regardless of $\unicode[STIX]{x1D70F}$. (b,c) Forcings ${\mathcal{C}}_{w}$ (grey lines) and ${\mathcal{S}}_{w}$ (black lines) for (b) $F_{h}=0.1$ and (c) $F_{h}=0.5$, at $z=l_{z}/4$.

In the next two sections, we will determine how the angular velocity evolves on the slow time $\unicode[STIX]{x1D70F}$. Then, (3.52) and (3.53) will be solved in §4.2.

3.2.2 Determination of $(\unicode[STIX]{x1D6F7}_{10},\unicode[STIX]{x1D713}_{10})$

We now determine $(\unicode[STIX]{x1D6F7}_{10},\unicode[STIX]{x1D713}_{10})$ corresponding to the azimuthal mode $m=0$ of the potential and streamfunction at first order (3.32). In order to keep the calculations relatively simple, we will assume here that the Froude number is small so that the hydrostatic approximation is satisfied. The comparison with the DNS in Part 2 will show that this assumption provides accurate predictions at least for $F_{h}\leqslant 0.5$. We will therefore take the limit $F_{h}\rightarrow 0$ of the vertical velocity and buoyancy (3.23) and (3.24). In addition, equations (3.52) and (3.53) for the displacement perturbations derived in the previous section show that $(\unicode[STIX]{x1D6FF}x,\unicode[STIX]{x1D6FF}y)$ remain of order unity for large time $t=O(1/\unicode[STIX]{x1D700})$ and are therefore small compared to the terms growing like $t$. Therefore, these displacements will be neglected in (3.23) and (3.24) in order to avoid lengthy calculations. Hence, (3.23) and (3.24) are simplified as follows:

(3.55)$$\begin{eqnarray}\displaystyle & \displaystyle w_{0}=\left[\cos (\tilde{\unicode[STIX]{x1D703}})-t\sin (\tilde{\unicode[STIX]{x1D703}})\unicode[STIX]{x1D6FA}-\cos (\unicode[STIX]{x1D6FA}t-\tilde{\unicode[STIX]{x1D703}})\cos \left(\frac{t}{F_{h}}\right)\right]\tilde{r}\unicode[STIX]{x1D6FA}^{2}\frac{\text{d}U}{\text{d}\tilde{z}}, & \displaystyle\end{eqnarray}$$

(3.56)$$\begin{eqnarray}\displaystyle & \displaystyle b_{0}=-t\cos (\tilde{\unicode[STIX]{x1D703}})\tilde{r}\unicode[STIX]{x1D6FA}^{2}\frac{\text{d}U}{\text{d}\tilde{z}}. & \displaystyle\end{eqnarray}$$

The only remaining term due to internal waves is the third term in (3.55). The forcing terms in (3.33) and (3.34) for $m=0$ and $m=2$, i.e. $(\unicode[STIX]{x1D6FF}D_{0},\unicode[STIX]{x1D6FF}D_{2},\unicode[STIX]{x1D6FF}Z_{0},\unicode[STIX]{x1D6FF}Z_{2})$, have been derived using (3.55).

Equation (3.33) for the potential $\unicode[STIX]{x1D6F7}_{10}$ reads

(3.57)$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6FB}_{h}^{2}\unicode[STIX]{x1D6F7}_{10}=\left[\frac{\unicode[STIX]{x2202}\tilde{r}\unicode[STIX]{x1D6FA}}{\unicode[STIX]{x2202}\tilde{r}}-\left(\frac{\unicode[STIX]{x2202}\tilde{r}\unicode[STIX]{x1D6FA}}{\unicode[STIX]{x2202}\tilde{r}}\cos (\unicode[STIX]{x1D6FA}t)-\frac{t}{2}\tilde{r}\unicode[STIX]{x1D6FA}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D6FA}}{\unicode[STIX]{x2202}\tilde{r}}\sin (\unicode[STIX]{x1D6FA}t)\right)\cos \left(\frac{t}{F_{h}}\right)\right]\unicode[STIX]{x1D6FA}t\left(\frac{\text{d}U}{\text{d}\tilde{z}}\right)^{2}. & & \displaystyle \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

The solution, which is not singular at $\tilde{r}=0$ and which vanishes at infinity, is

(3.58)$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6F7}_{10}=\frac{t}{2}\left[\int _{+\infty }^{\tilde{r}}\unicode[STIX]{x1D709}\unicode[STIX]{x1D6FA}^{2}\,\text{d}\unicode[STIX]{x1D709}-\cos \left(\frac{t}{F_{h}}\right)\int _{+\infty }^{\tilde{r}}\unicode[STIX]{x1D709}\unicode[STIX]{x1D6FA}^{2}\cos (\unicode[STIX]{x1D6FA}t)\,\text{d}\unicode[STIX]{x1D709}\right]\left(\frac{\text{d}U}{\text{d}\tilde{z}}\right)^{2}. & & \displaystyle\end{eqnarray}$$

Using (3.58), equation (3.34) for the streamfunction $\unicode[STIX]{x1D713}_{10}$ is

(3.59)$$\begin{eqnarray}\displaystyle \text{L}(\unicode[STIX]{x1D713}_{10})=-\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D701}_{0}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D70F}}+{\mathcal{F}}_{10}, & & \displaystyle\end{eqnarray}$$

where

(3.60)$$\begin{eqnarray}\displaystyle & & \displaystyle {\mathcal{F}}_{10}=\left[-\frac{t}{\tilde{r}}\frac{\unicode[STIX]{x2202}\tilde{r}^{2}\unicode[STIX]{x1D6FA}^{3}}{\unicode[STIX]{x2202}\tilde{r}}-\left[\unicode[STIX]{x1D6FA}\left(\frac{\unicode[STIX]{x2202}\tilde{r}\unicode[STIX]{x1D6FA}}{\unicode[STIX]{x2202}\tilde{r}}+\frac{t^{2}}{2}\tilde{r}\unicode[STIX]{x1D6FA}^{2}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D6FA}}{\unicode[STIX]{x2202}\tilde{r}}\right)\sin (\unicode[STIX]{x1D6FA}t)\right.\right.\nonumber\\ \displaystyle & & \displaystyle \quad \left.\left.-\;t\unicode[STIX]{x1D6FA}^{2}\frac{\unicode[STIX]{x2202}\tilde{r}\unicode[STIX]{x1D6FA}}{\unicode[STIX]{x2202}\tilde{r}}\cos (\unicode[STIX]{x1D6FA}t)\right]\cos \left(\frac{t}{F_{h}}\right)\right.+\left.\frac{\tilde{\unicode[STIX]{x1D70F}}^{2}}{2\tilde{Re}_{b}}\frac{1}{\tilde{r}}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\tilde{r}}\left(\tilde{r}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D701}_{0}}{\unicode[STIX]{x2202}\tilde{r}}\right)\right]\left(\frac{\text{d}U}{\text{d}\tilde{z}}\right)^{2},\end{eqnarray}$$

where we remind the reader that $\tilde{\unicode[STIX]{x1D70F}}=\unicode[STIX]{x1D700}t$. Since $\unicode[STIX]{x1D713}_{10}$ is axisymmetric, the operator $\text{L}$ in (3.59) reduces to $\unicode[STIX]{x2202}\unicode[STIX]{x1D6FB}_{h}^{2}\unicode[STIX]{x1D713}_{10}/\unicode[STIX]{x2202}t$. In addition, we should impose that

(3.61)$$\begin{eqnarray}\left\langle \unicode[STIX]{x1D6FB}_{h}^{2}\unicode[STIX]{x1D713}_{10},\unicode[STIX]{x1D701}_{0}\right\rangle =0,\end{eqnarray}$$

in order that the vertical vorticity at first order is orthogonal to the vertical vorticity at leading order. Taking the scalar product of (3.59) with $\unicode[STIX]{x1D701}_{0}$ shows that $\unicode[STIX]{x2202}\langle \unicode[STIX]{x1D6FB}_{h}^{2}\unicode[STIX]{x1D713}_{10},\unicode[STIX]{x1D701}_{0}\rangle /\unicode[STIX]{x2202}t=0$ only if

(3.62)$$\begin{eqnarray}\displaystyle \frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D701}_{0}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D70F}}={\mathcal{F}}_{10}. & & \displaystyle\end{eqnarray}$$

This equation describes the evolution of the leading-order vertical vorticity of the vortex (and associated angular velocity) over the slow time. It is analysed in § 4.1. The potential and streamfunction $(\unicode[STIX]{x1D6F7}_{12},\unicode[STIX]{x1D713}_{12})$ are solved in appendix D.