2.1 Structure of eddy diffusivity

When fluid motions are turbulent, it is common practice to divide the velocity into mean and fluctuating parts, and consider a set of equations governing the spatial structure of the mean flow, with the effect of small-scale motions parameterized in some manner. A simple and reasonably effective approximation is to quantify the effect of small-scale motions using an eddy diffusivity, $\unicode[STIX]{x1D708}$, e.g. see § 23.5.2 of Loper (Reference Loper2017). Of course, the effectiveness of this approach depends on the manner in which $\unicode[STIX]{x1D708}$ is parameterized.

The eddy diffusivity has dimensions of length squared divided by time, or equivalently speed times length. Since $\unicode[STIX]{x1D708}$ encapsulates the effect of small-scale turbulent motions induced by a velocity difference (quantified by $v_{\infty }$ in the present case), it is reasonable to suppose that $\unicode[STIX]{x1D708}$ depends on $v_{\infty }$. Dimensional consistency requires that this relation be linear, leading directly to the parameterization

(2.8)$$\begin{eqnarray}\unicode[STIX]{x1D708}(r,z)=v_{\infty }(r)L(r,z),\end{eqnarray}$$

where the diffusivity function $L(r,z)$ has dimensions of length; see § 4.

Usually, within the context of mixing-length theory, the eddy diffusivity is expressed as the product of a local velocity gradient times the square of the mixing length; e.g. see § 5.3.3 of Holton (Reference Holton2004) or § 23.5 of Loper (Reference Loper2017). Since the local velocity gradient is proportional to $v_{\infty }$, this gradient may be replaced by $v_{\infty }$ divided by a gradient scale. The diffusivity function $L$ is in effect a modified mixing length equal to the square of the mixing length divided by that gradient scale. Equation (2.8) is structurally identical to formula (19.9) of Schlichting (Reference Schlichting1968), except that here $L$ may be a function of $r$ and $z$, whereas the corresponding factor ($\unicode[STIX]{x1D712}_{1}b$) is constant in Schlichting’s version. It is shown in § 23.5.2 of Loper (Reference Loper2017) that $L\ll z$.

The dynamic balance within a boundary layer is between inertia and a viscous force that may be directly due to molecular viscosity or may be a representation of small-scale inertia. A consequence of formulation (2.8) is that the viscous force representing small-scale inertia now has the same mathematical structure as the large-scale inertial terms: both are quadratic in $v_{\infty }$. This similarity of structure underlies the versatility of the formulation developed in § 3. There are many other, more sophisticated models of eddy diffusivity (e.g. see Fiedler & Garfield (Reference Fiedler and Garfield2010)), but they lack the elegant structure of (2.8) that mimics the form of the inertial terms and thus permits development of governing equations that are independent of the magnitude of $v_{\infty }$.

3.1 Complex formulation

As noted by Kuo (Reference Kuo1971), the two momentum equations (3.5) and (3.6) have similar structure and are readily combined into a single complex equation. Writing

(3.12)$$\begin{eqnarray}\boldsymbol{V}=G+\boldsymbol{i}F,\end{eqnarray}$$

these equations may be combined into

(3.13)$$\begin{eqnarray}(\unicode[STIX]{x1D6EC}\boldsymbol{V}_{\unicode[STIX]{x1D701}})_{\unicode[STIX]{x1D701}}-H\boldsymbol{V}_{\unicode[STIX]{x1D709}}-\unicode[STIX]{x1D70C}F\boldsymbol{V}_{\unicode[STIX]{x1D70C}}+2(1-\unicode[STIX]{x1D703})F\boldsymbol{V}+\boldsymbol{i}\boldsymbol{V}^{2}=\boldsymbol{i}.\end{eqnarray}$$

Note that, here and in the following, complex quantities (including the imaginary unit $\boldsymbol{i}$) are expressed using bold italic letters.

Conditions (3.11) now become

(3.14)$$\begin{eqnarray}\boldsymbol{V}(\unicode[STIX]{x1D70C},0)=H(\unicode[STIX]{x1D70C},0)=\boldsymbol{V}(\unicode[STIX]{x1D70C},\infty )-1=0.\end{eqnarray}$$

3.2 Discussion of the non-dimensionalization

The non-dimensionalization employs non-orthogonal independent variables; while $\unicode[STIX]{x1D70C}$ is a dimensionless radial coordinate, $\unicode[STIX]{x1D701}$ depends on both $z$ and $r$. This choice of independent variables serves to minimize the effect of radial variation. In particular, if the parameters $\unicode[STIX]{x1D703}$ and $\unicode[STIX]{x1D6EC}$ are independent of $\unicode[STIX]{x1D70C}$, then that variable does not appear explicitly in (3.7) and (3.13) or in conditions (3.14). Consequently, the solution is independent of $\unicode[STIX]{x1D70C}$; the radial derivatives in these equations may be ignored, making them ordinary differential equations. In other words, if $\unicode[STIX]{x1D703}$ and $\unicode[STIX]{x1D6EC}$ are independent of $\unicode[STIX]{x1D70C}$, then the non-dimensionalization is a similarity transform. This is the case for diffusivity model A introduced in § 4, having the function $L$ equal to a constant. However, diffusivity model B has $L$ varying with $z$. Since $z=z_{0}\unicode[STIX]{x1D701}\sqrt{\unicode[STIX]{x1D70C}}$, $\unicode[STIX]{x1D6EC}$ depends on $\unicode[STIX]{x1D70C}$ and the radial-derivative terms must be retained; when model B is employed, the transformation is not a similarity.

The equations governing rotating flows with $\unicode[STIX]{x1D708}$ assumed constant have been previously reduced to ordinary differential equations using

1. (i) the von Kármán–Bödewadt transformation (for rigid-body rotation with $v_{\infty }\sim r$) having $\unicode[STIX]{x1D701}$ independent of $r$;

2. (ii) Long’s conical transformation (for a potential vortex with $v_{\infty }\sim 1/r$) having $\unicode[STIX]{x1D701}\propto z/r$; and

3. (iii) Kuo’s power-law transformation (for power-law vortex with $v_{\infty }\sim r^{2n-1}$) having $\unicode[STIX]{x1D701}\propto r^{n-1}z$;

(von Kármán (Reference von Kármán1921), Bödewadt (Reference Bödewadt1940), Long (Reference Long1958), Kuo (Reference Kuo1971); see also Foster (Reference Foster2009) and Bĕlík et al. (Reference Bĕlík, Dokken, Schloz and Shvartsman2014)). Note that Kuo’s transformation is a hybrid of the first two, equalling the Kármán–Bödewadt similarity transformation when $n=1$ and the Long transformation when $n=0$.

In each of these cases, the structure of the similarity transformation is keyed to a specific radial variation of the flow outside the boundary layer. The transformation given by (3.1)–(3.3) appears to be novel, in that there is no comparable restriction on the form of $v_{\infty }$. With $\unicode[STIX]{x1D708}\propto v_{\infty }(r)$, the structure of the principal transform variable $\unicode[STIX]{x1D701}\propto z/\sqrt{r}$ is independent of the radial variation of the outer flow and the function $v_{\infty }(r)$ enters the formulation parametrically, through the swirl parameter, $\unicode[STIX]{x1D703}$. This feature is important because it permits consideration of a large class of swirling flows. This flexibility in the formulation permits the axial component of velocity to be of smaller order than the radial component, in contrast to Long’s conclusion that the two must be of the same order of magnitude. The difference between present formulation and previous studies that treated the eddy diffusivity as constant is seen most clearly in the continuity equation; if the diffusivity were constant (as in the laminar Bödewadt problem considered in appendix C), the factor multiplying $F$ in the continuity equation would be 2 rather than $2\unicode[STIX]{x1D703}+1/2$.

The present formulation is mathematically identical to the Blasius transformation, quantifying flow over a semi-infinite flat plate (see § VII.e, pp. 125–133 of Schlichting (Reference Schlichting1968)), with the horizontal coordinate ($r$) being distance from the leading edge in the Blasius problem and distance from the vortex axis in the present problem. As with the Blasius transformation, in the present formulation the horizontal velocity components are not explicitly dependent on $r$, while both $\unicode[STIX]{x1D701}$ and the axial velocity vary explicitly as $1/\sqrt{r}$, so that the solution has a weak algebraic singularity as $r\rightarrow 0$. This weak singularity governs the behaviour of the solution in the limit $r\rightarrow 0$ for rigid-body outer flow, but plays no role in the structure of the boundary layer for vortical outer flow (having $\unicode[STIX]{x1D703}<0.5$); it will be seen in Part 2 that when the outer flow is vortical, the boundary-layer formulation breaks down at a finite radius.

The structure of the outer flow and its effect on the boundary layer is encapsulated in the parameter $\unicode[STIX]{x1D703}$ defined by (3.9), which is a dimensionless measure of both the angular-momentum gradient and vorticity outside the boundary layer. Specifically, the axial vorticity $\unicode[STIX]{x1D714}$ of the outer flow is related to $\unicode[STIX]{x1D703}$ by

(3.15)$$\begin{eqnarray}\unicode[STIX]{x1D714}=\frac{1}{r}\frac{\text{d}(rv_{\infty })}{\text{d}r}=2\unicode[STIX]{x1D703}\frac{v_{\infty }}{r}.\end{eqnarray}$$

Both $\unicode[STIX]{x1D714}$ and $\unicode[STIX]{x1D703}$ quantify the ‘stiffness’ of the outer flow, as exemplified by its ability to resist radial motion and sustain inertial waves. This stiffness causes the axial oscillation of the boundary-layer variables, with viscosity causing an axial damping.

If $\unicode[STIX]{x1D703}$ is constant, the outer flow is a simple power-law flow with

(3.16)$$\begin{eqnarray}v_{\infty }\sim r^{2\unicode[STIX]{x1D703}-1}.\end{eqnarray}$$

Note that if

1. (I) $\unicode[STIX]{x1D703}<0$, the angular-momentum gradient is negative and outer flow is dynamically unstable;

2. (II) $\unicode[STIX]{x1D703}=0$, the outer flow is a potential vortex;

3. (III) $0<\unicode[STIX]{x1D703}<1$, the outer flow is a general swirling motion and more specifically

   1. (i) for $0<\unicode[STIX]{x1D703}<1/2$, $v_{\infty }$ is vortical, being a decreasing function of $r$ and infinite at $r=0$;

   2. (ii) for $\unicode[STIX]{x1D703}=1/2$, $v_{\infty }$ is independent of $r$ and finite (and singular) at $r=0$; and

   3. (iii) for $1/2<\unicode[STIX]{x1D703}<1$, $v_{\infty }$ is rotational: being an increasing function of $r$ and zero at $r=0$;

4. (IV) $\unicode[STIX]{x1D703}=1$, the outer flow is rigid-body rotation and the problem becomes a turbulent version of the classic Bödewadt problem; this is investigated in §§ 5 and 6;

5. (V) $1<\unicode[STIX]{x1D703}$, the outer flow is ‘super rigid-body’ rotation, with the circumferential velocity increasing with $r$ faster than linear.

The flow in an atmospheric vortex has $0<\unicode[STIX]{x1D703}<1$ in the region outside the eyewall and $\unicode[STIX]{x1D703}>1$ within the eye, with a transition within the eyewall.

The singular behaviour of the boundary-layer formulation in the limit $\unicode[STIX]{x1D703}\rightarrow 0$ (that is, swirl tending to a potential vortex) is seen in the circumferential momentum equation (3.6); radial advection of angular momentum goes to zero in this limit and there is no mechanism to balance its axial diffusion in a steady state. This singular behaviour is fundamental and cannot be eliminated by re-scaling the variables. As $\unicode[STIX]{x1D703}$ decreases toward 0, the outer fluid becomes increasingly flaccid; the boundary layer thickens and the meridional flow speed increases. Many studies of swirling flow have assumed the outer flow to be a potential vortex, with the implicit assumption that such flow is representative of vortical flows. However, this is not the case; potential flow is in fact a singular limit of a general swirling flow. It is known (Goldshtik Reference Goldshtik1990) that for laminar flow, the boundary-layer problem describing flow beneath a potential vortex has a solution only if the Reynolds number $rv_{\infty }/\unicode[STIX]{x1D708}$ is less than 5.53. This situation also occurs when the flow is turbulent, with no solution to the boundary-layer problem if the outer flow is a potential vortex, or even close to it. However, with a solution procedure in hand, it is possible to follow in some detail the behaviour of the solution as this singular limit is approached. This investigation is beyond the scope of this article; it is taken up in Part 2.

3.2.1 Limitation of boundary-layer formulation

Two assumptions are fundamental to the boundary-layer approach to finding the flow near a bounding surface: axial derivatives are much larger than transverse derivatives and the axial velocity is small compared with the transverse velocity components. In the present case these assumptions are satisfied provided

(3.17)$$\begin{eqnarray}\unicode[STIX]{x2202}/\unicode[STIX]{x2202}r\ll \unicode[STIX]{x2202}/\unicode[STIX]{x2202}z\end{eqnarray}$$

and $w\ll v_{\infty }$ or equivalently

(3.18)$$\begin{eqnarray}\unicode[STIX]{x1D716}H^{\star }\ll 1.\end{eqnarray}$$

With the former assumption the radial viscous terms may be neglected and with the latter assumption, the axial momentum equation reduces to $\unicode[STIX]{x2202}p/\unicode[STIX]{x2202}z=0$ to dominant order, with the external pressure field being impressed on the boundary layer, as is done in § 2. Often these conditions go hand in hand; that is, they are simultaneously satisfied or not.

If $F$ and $H$ are of unit order, it follows from (3.8) that $H^{\star }=O(\unicode[STIX]{x1D70C}^{-1/2})$, and (3.18) is satisfied provided that

(3.19)$$\begin{eqnarray}\unicode[STIX]{x1D716}z_{0}\ll r.\end{eqnarray}$$

This limitation is similar to that for the Blasius boundary layer, which is not valid close to the leading edge of the flat plate. This restriction on the domain of validity shows that a boundary layer cannot exist close to the axis of a vortical flow; flow in that region must be different from that investigated in this study.

If $H$ is of unit order, as it is for $\unicode[STIX]{x1D703}=1$, condition (3.19) is not a severe restriction on the domain of validity of the boundary-layer solution. However, if $H$ becomes large, as it does for $\unicode[STIX]{x1D703}<0.5$, condition (3.18) places a limit on the radial domain in which the boundary-layer formalism is valid. That is, it is seen in Part 2 that, for $\unicode[STIX]{x1D703}<0.42$, condition (3.18) fails at a finite radius, which presumably signifies the presence of an eye and corner region.

The problem consists of equations (3.7) and (3.13) to be solved for $H$, $\boldsymbol{V}$ and $F=\text{Im}[\boldsymbol{V}]$ on the interval $0<\unicode[STIX]{x1D701}<\infty$, subject to conditions (3.14). Equation (3.13) contains one parameter $\unicode[STIX]{x1D703}$, defined by (3.9), and a diffusivity function $\unicode[STIX]{x1D6EC}$ that is defined in the following section. If the outer flow is a simple power of $r$ then $\unicode[STIX]{x1D703}$ is a constant. This leads to the investigation of a sequence of problems, beginning with rigid-body motion $\unicode[STIX]{x1D703}=1$ in the following sections of this paper. Part 2 considers the structure of the boundary layer beneath a power-law swirl, for which $\unicode[STIX]{x1D703}$ is constant.

4.1 Diffusivity model A

The simplest model of the diffusivity function is

(4.1)$$\begin{eqnarray}L=\unicode[STIX]{x1D716}z_{0},\end{eqnarray}$$

where $\unicode[STIX]{x1D716}$ and $z_{0}$ are constants. It is readily seen that (3.10) simplifies to

(4.2)$$\begin{eqnarray}\unicode[STIX]{x1D6EC}=1.\end{eqnarray}$$

This model is a parameterization of the resistance to turbulent flow afforded by a homogeneous array of small-scale obstacles; it is essentially a model of turbulent flow within a porous medium. This model is generalized and made more geophysically plausible in the following subsection.

4.2 Diffusivity model B

The physical basis for model B is the realization that the lower boundary for atmospheric vortices is not smooth. Turbulence is induced near the ground by an array of blunt obstacles (e.g. topographic irregularities, vegetation, buildings, etc.) within a rough layer that extends a distance $z_{0}$ from the ground. These obstacles act as literal trip wires that induce turbulence within and above the layer. Within the rough layer uniform resistance, with $L=\unicode[STIX]{x1D716}z_{0}$, is retained. Above the rough layer, turbulence is maintained by shear-flow instability. Representation of the diffusivity function in the region above the layer is based on Prandtl’s mixing-length theory, with $\unicode[STIX]{x1D708}=\unicode[STIX]{x1D716}v_{\infty }z$. This is equivalent to formula (19.9) of Schlichting (Reference Schlichting1968; see also § 23.5.2 of Loper (Reference Loper2017)); that is, $L=\unicode[STIX]{x1D716}z$ above the rough layer. Altogether,

(4.3)$$\begin{eqnarray}L=\unicode[STIX]{x1D716}\left\{\begin{array}{@{}ll@{}}z_{0} & \text{if}~0<z\leqslant z_{0}\\ z & \text{if}~z_{0}<z\end{array}\right.\end{eqnarray}$$

for model B. Substituting (4.3) into (3.10) the dimensionless diffusivity function becomes

(4.4)$$\begin{eqnarray}\unicode[STIX]{x1D6EC}=\left\{\begin{array}{@{}ll@{}}1 & \text{if}~0<\unicode[STIX]{x1D701}\leqslant 1/\sqrt{\unicode[STIX]{x1D70C}}\\ \unicode[STIX]{x1D701}\sqrt{\unicode[STIX]{x1D70C}} & \text{if}~1/\sqrt{\unicode[STIX]{x1D70C}}<\unicode[STIX]{x1D701}.\end{array}\right.\end{eqnarray}$$

The parameter $\unicode[STIX]{x1D70C}$, defined by (3.2), may be viewed in two ways. With $z_{0}$ constant, $\unicode[STIX]{x1D70C}$ is the non-dimensional radius. On the other hand, at a given radius, $\unicode[STIX]{x1D70C}$ is a measure of the effect on the boundary-layer flow of a rough layer having thickness $z_{0}$. Note that:

1. (i) in the limit $\unicode[STIX]{x1D70C}\rightarrow 0$ the mathematical problem using model B approaches that using model A; however, the physical interpretation of the results differ for the two models; results using model A apply for all values of $r$, while using model B, results in the limit $\unicode[STIX]{x1D70C}\rightarrow 0$ apply only as $r\rightarrow 0$ – not for all $r$;

2. (ii) in the limit $\unicode[STIX]{x1D70C}\rightarrow \infty$ the resistive influence of the rough layer tends to zero and model B becomes singular;

3. (iii) with $z_{0}$ constant, the relative importance of the rough layer varies as $1/\sqrt{r}$; that is, the effect of the rough layer is relatively small far from the axis of rotation and increases as $r\rightarrow 0$; and

4. (iv) this formulation of the diffusivity function above the rough layer is in accord with the line of reasoning found in Bak (Reference Bak1996) and the idea that the mean shearing motions within the boundary layer are at the margin of stability, with the local Reynolds number nearly constant

   (4.5)$$\begin{eqnarray}Re=\frac{v_{\infty }z}{\unicode[STIX]{x1D708}}=\frac{1}{\unicode[STIX]{x1D716}}.\end{eqnarray}$$

This Reynolds number is likely to be moderately larger than the value at which vortex shedding about a cylindrical obstacle first occurs: $Re\approx 47$. It follows that $\unicode[STIX]{x1D716}<0.02$. (Loper (Reference Loper2017, § 23.5.2) arrived at this estimated value in a different manner, with $\unicode[STIX]{x1D716}=\unicode[STIX]{x1D705}\sqrt{C_{D}}$, where $\unicode[STIX]{x1D705}=0.421$ is the von Kármán constant and $C_{D}\approx 0.0013$ is drag coefficient.) The precise value of $\unicode[STIX]{x1D716}$ is not important. The only requirement is that it is much less than unity, so that the neglect of the radial viscous terms, which are of order $\unicode[STIX]{x1D716}^{2}$, is justified.

With $\unicode[STIX]{x1D716}\approx 0.01$, say, and $z_{0}$ ranging from 0.1 m (grassland) to 10 m (forest) and $r$ ranging from 10 m (for a dust devil) to 1000 m (for a tornado), a plausible range for $\unicode[STIX]{x1D70C}$ is $0.01<\unicode[STIX]{x1D70C}<100.0$. Condition (3.19) requires that $\unicode[STIX]{x1D716}^{2}\ll \unicode[STIX]{x1D70C}$; this condition is well satisfied with these parameter estimates.

This parameterization, with diffusivity growing with $z$, succeeds in giving reasonable answers because inertial stability of the far fluid causes the boundary-layer variables to decay exponentially with axial distance. If the far fluid is not rotating (as for example in the von Kármán problem) this simple parameterization fails. Finally, it should be noted that many other parameterizations of the diffusivity function could be used. The choice will affect numerical values of the results presented in §§ 5 and 6, but it is believed that the qualitative character of the solutions will be unchanged.

4.2.1 The rough layer and the slip condition

A diffusivity function that varies linearly with $z$ is associated with a velocity profile that is logarithmic near the boundary (e.g. see § 5.3.5 of Holton (Reference Holton2004) and/or § 23.6 of Loper (Reference Loper2017)) and normally the boundary-layer problem requires specification of a slip boundary condition at $z=0$. The slip condition is a relation between the velocity and its normal gradient containing a drag coefficient; e.g. see (3.10) of Eliassen (Reference Eliassen1971). The drag coefficient represents (parameterizes) the effect of small-scale motions over and around tiny irregularities on the surface; e.g. see §§ 23.4 and 23.5 of Loper (Reference Loper2017). When considering atmospheric vortices, this simple parameterization is unsatisfactory because the so-called small-scale irregularities (that is, trees, buildings. etc) are not that small, and considerable flow occurs within this rough layer. (The need to explicitly resolve the flow within the rough layer is clearly evident from the left-hand panels of figures 8, 9 and 10. Note that the rough layer extends from $z^{\star }=0$ to $z^{\star }=1$. In particular, the left-hand panel of figure 8 shows that all of the primary jet lies within the rough layer if $\unicode[STIX]{x1D70C}\leqslant 0.1$.) This realization led to the development of a more sophisticated model of the interaction of turbulent flow with a rough boundary in which the layer containing the roughness elements is of finite size, with flow within this layer being explicitly resolved. In this model, drag is produced by the macroscopic obstacles within the rough layer, rather than by microscopic irregularities close to the ground. Essentially, the rough layer is a porous medium within which flow is turbulent. Since the drag is represented by a body force in this model, there is no need to introduce the heuristic slip condition and the more-precise no-slip condition of laminar models may be retained. This results in a well-posed problem provided $z_{0}>0$. The momentum equation is singular in the limit $z_{0}\rightarrow 0$, or equivalently $\unicode[STIX]{x1D70C}\rightarrow \infty$, with velocity profiles near the boundary approaching the well-known logarithmic shape (see the right-hand panels of figures 8 and 9).

For the remainder of this article, attention is confined to Bödewadt flow, with the far fluid being in rigid-body motion with

(4.6)$$\begin{eqnarray}v_{\infty }(r)=\unicode[STIX]{x1D6FA}r,\end{eqnarray}$$

where $\unicode[STIX]{x1D6FA}$ is the rotation rate. In this case $\unicode[STIX]{x1D703}=1$. The solution to the problem formulated in §§ 3 and 4 with $\unicode[STIX]{x1D703}=1$ using model A is investigated in the following section, while the solution using model B is presented in § 6.

5.1 Comparison of model A and the laminar Bödewadt model

Model A is similar to the laminar Bödewadt model, in that the diffusivity is independent of axial distance in both models. In fact the momentum equations for these two models are mathematically identical; compare (5.2) with (C 8). However, the models are not physically identical, because the continuity equations and meanings of the quantity $H$ differ. In Bödewadt flow, lines of constant $\unicode[STIX]{x1D701}$ are parallel to lines of constant $z$ and, as a consequence, $H$ is both the axial speed and the speed normal to lines (in the meridional plane) of constant $\unicode[STIX]{x1D701}$. With the turbulent non-dimensionalization, lines of constant $\unicode[STIX]{x1D701}$ are tilted relative to lines of constant $z$ and consequently there is a difference between $H$, which is the speed normal to lines of constant $\unicode[STIX]{x1D701}$, and $H_{A}$, which is the speed normal to lines of constant $z$; see (5.3). The difference between $H$ and $H_{A}$ is visualized in the inset of figure 12. This distinction between the physical meanings shows up in the continuity equations, which is $H^{\prime }+(5/2)F=0$ for the turbulent Bödewadt model and $H^{\prime }+2F=0$ for the laminar Bödewadt model; compare (5.1) with (C 5). A physical consequence of this difference is that meridional flow is larger for the turbulent model, as illustrated by the hodographs in figure 3.

Another difference between the laminar and turbulent Bödewadt solutions is in the relation between dimensional and dimensionless axial velocity components. In the laminar case, these two are related by a constant factor $\sqrt{\unicode[STIX]{x1D708}\unicode[STIX]{x1D6FA}}$, while in the turbulent case, they are related by a factor that depends on $r$ (see (3.1) and (3.8)); the dimensional turbulent axial velocity varies as $1/\sqrt{r}$. This radial dependence, which is implicit in the solution using model A, is made explicit when investigating the structure of the solution using model B.

6.1 Visualization of results

When visualizing results obtained using model B the appropriate scaling of the radial and axial coordinates is $\unicode[STIX]{x1D70C}$ and $z^{\star }$ defined by (3.2) and (3.4), respectively. Also, using (3.1), (3.3) and (3.8) the velocity components and their gradients may be expressed as

(6.1)$$\begin{eqnarray}\displaystyle \left.\begin{array}{@{}c@{}}\displaystyle \frac{u}{v_{\infty }}=F,\quad \frac{v}{v_{\infty }}=G,\quad \frac{w}{\unicode[STIX]{x1D716}v_{\infty }}=H^{\star },\\ \displaystyle \frac{z_{0}}{v_{\infty }}\frac{\unicode[STIX]{x2202}u}{\unicode[STIX]{x2202}z}=\frac{F^{\prime }}{\sqrt{\unicode[STIX]{x1D70C}}}\quad \text{and}\quad \frac{z_{0}}{v_{\infty }}\frac{\unicode[STIX]{x2202}v}{\unicode[STIX]{x2202}z}=\frac{G^{\prime }}{\sqrt{\unicode[STIX]{x1D70C}}}.\end{array}\right\} & & \displaystyle\end{eqnarray}$$

With this in mind,

1. (i) values of $z_{1}^{\star }$, $z_{2}^{\star }$, $z_{G}^{\star }$ and $z_{F}^{\star }$ are graphed versus $\unicode[STIX]{x1D70C}$ in figure 4;

2. (ii) values of $H_{\infty }^{\star }$ and $H_{max}^{\star }$ are graphed versus $\unicode[STIX]{x1D70C}$ in figure 5;

3. (iii) values of $F_{min}$ and $G_{max}$ are graphed versus $\unicode[STIX]{x1D70C}$ in figure 6; and

4. (iv) values of $F^{\prime }/\sqrt{\unicode[STIX]{x1D70C}}$ and $G^{\prime }/\sqrt{\unicode[STIX]{x1D70C}}$ evaluated at $\unicode[STIX]{x1D701}=0$ are graphed versus $\unicode[STIX]{x1D70C}$ in figure 7.

Each of these figures consists of two panels, with graphs in the right-hand panels extending from $\unicode[STIX]{x1D70C}=0$ to 100 and those the left-hand panels extending from $\unicode[STIX]{x1D70C}=0$ to 1.5, in order to show more clearly the structure close to the axis of rotation.

Figure 4. Graphs of the dimensionless axial locations of the first ($z_{1}^{\star }$) and second ($z_{2}^{\star }$) zeros of the radial component of velocity, the maximum ($z_{G}^{\star }$) of the circumferential component of velocity and the minimum ($z_{F}^{\star }$) of the radial component of velocity using model B: for $0<\unicode[STIX]{x1D70C}<100$ in (b) and for $0<\unicode[STIX]{x1D70C}\lesssim 1.5$ in (a). The solid curves delimit the domains of radial flow, with the double arrows denoting the direction of flow in the meridional plane. The radial inflow within $0<z^{\star }<z_{1}^{\star }$ is the primary jet and the radial outflow within $z_{1}^{\star }<z^{\star }<z_{2}^{\star }$ is the secondary jet.

Figure 5. Graphs of the asymptotic ($H_{\infty }^{\star }$) and maximum ($H_{max}^{\star }$) values of the axial velocity component: for $0<\unicode[STIX]{x1D70C}<100$ in (b) and for $0<\unicode[STIX]{x1D70C}\lesssim 1.5$ in (a).

Figure 6. Graphs of the maximum circumferential ($G_{max}$) and minimum radial ($F_{min}$) velocity components: for $0<\unicode[STIX]{x1D70C}<100$ in (b) and for $0<\unicode[STIX]{x1D70C}\lesssim 1.5$ in (a).

6.2 Boundary-layer shape

The shape of the boundary layer is illustrated in figure 4, which contains graphs of the dimensionless axial locations $z_{1}^{\star }$, $z_{2}^{\star }$, $z_{G}^{\star }$ and $z_{F}^{\star }$ versus $\unicode[STIX]{x1D70C}$. It is seen that overall the boundary-layer thickness varies roughly linearly with radius, tending to zero at the axis of rotation. There is a departure from linearity for $\unicode[STIX]{x1D70C}<1$, with the layer thickness tending toward (but not achieving) the parabolic shape of model A. The variation with $\unicode[STIX]{x1D70C}$ of the axial locations seen in this figure may be parameterized by

(6.2)$$\begin{eqnarray}z_{y}^{\star }=a_{y}\sqrt{\unicode[STIX]{x1D70C}}+b_{y}\unicode[STIX]{x1D70C}\end{eqnarray}$$

for $y=1$, 2, $G$ and $F$, with $a_{1}=2.018$, $b_{1}=2.171$, $a_{2}=4.391$, $b_{2}=9.256$, $a_{G}=2.009$, $b_{G}=1.128$, $a_{F}=0.875$ and $b_{F}=0.034$.

The two solid curves in figure 4 visualize the axial locations of the first and second interior zeros of $F$. Within $0<z^{\star }<z_{1}^{\star }$ the flow is radially inward; this is the primary jet; the outward-flowing secondary jet occurs within $z_{1}^{\star }<z^{\star }<z_{2}^{\star }$. That is, the lower solid curve marks the top of the primary jet, while the secondary jet lies between these two solid curves, as depicted by the double arrows in the right-hand panel of this figure.

6.3 Velocity magnitudes

The asymptotic ($H_{\infty }^{\star }$) and maximum ($H_{max}^{\star }$) values of the axial velocity component are graphed versus $\unicode[STIX]{x1D70C}$ in figure 5. The variation of these velocity components is governed primarily by the factor $\sqrt{\unicode[STIX]{x1D70C}}$ appearing in the definition of $H^{\star }$ (see (3.8)), with the speeds becoming arbitrarily large as $\unicode[STIX]{x1D70C}\rightarrow 0$. For large $\unicode[STIX]{x1D70C}$, $H_{\infty }^{\star }$ is moderately smaller than $H_{max}^{\star }$, indicating that oscillations in the meridional flow are weak. For small $\unicode[STIX]{x1D70C}$, $H_{max}^{\star }$ is considerably larger than $H_{\infty }^{\star }$, indicative of stronger oscillations.

The extreme values of $F$ and $G$ versus $\unicode[STIX]{x1D70C}$ are graphed in figure 6. They show little variation, with extremes having greatest magnitude for small $\unicode[STIX]{x1D70C}$.

6.4 Gradients at the boundary

The circumferential and radial shear stresses at the boundary are proportional to the gradients of the respective horizontal velocity components; see (6.1). The variation of these gradients with $\unicode[STIX]{x1D70C}$ is shown in figure 7. As with the axial velocity, the magnitude of the gradients is dominated by the factor $\sqrt{\unicode[STIX]{x1D70C}}$, becoming large as $\unicode[STIX]{x1D70C}$ decreases.

Figure 7. Graphs of the circumferential (positive values) and radial (negative values) boundary gradients: for $0<\unicode[STIX]{x1D70C}<100$ in (b) and for $0<\unicode[STIX]{x1D70C}\lesssim 1.5$ in (a).

6.5 Flow structure

The structure of the flow is illustrated in figures 8–10 by graphs of the radial ($F=u/v_{\infty }$), circumferential ($G=v/v_{\infty }$) and axial ($H^{\star }=\unicode[STIX]{x1D716}w/v_{\infty }$) components of velocity versus $z^{\star }$ for select values of $\unicode[STIX]{x1D70C}$: 0.1, 1.0, 10.0 and 100.0. The differences of vertical scale at these locations reflect the roughly linear shape of the boundary layer. For large $\unicode[STIX]{x1D70C}$, the boundary layer extends far above the top of the rough layer (at $z^{\star }=1$), while for $\unicode[STIX]{x1D70C}$ small, the boundary layer is comparable in height to the rough layer.

Figure 8. Graphs of the dimensionless radial velocity, $F$, versus $z^{\star }$ at four radial locations as indicated. The vertical dotted lines denote values of $F$ in one-tenth intervals. Note the changes in vertical scale.

Figure 9. Graphs of the dimensionless circumferential velocity, $G$, versus $z^{\star }$ at four radial locations as indicated. The vertical dotted lines denote the asymptotic value $G=1.0$. Note the changes in vertical scale.

Figure 10. Graphs of the dimensionless axial velocity, $H^{\star }$, versus $z^{\star }$ at four radial locations as indicated. The vertical dashed lines denote the asymptotic values of $H^{\star }$. The vertical dotted lines denote values of $H^{\star }$ in unit intervals. Note the changes in vertical scale.

All velocity components oscillate as $z$ increases, with $F$ changing sign, while $G$ and $H^{\star }$ remain positive. The strength of the oscillations increases as $\unicode[STIX]{x1D70C}$ decreases. The profiles of $F$ and $G$ show that for large $\unicode[STIX]{x1D70C}$, the boundary layer has a double-layer structure; the radial velocity has a maximum and $G$ surpasses unity at a modest value of $z^{\star }$, then $F$ relaxes to zero and the overshoot of $G$ diminishes over a large range of $z^{\star }$. For $z^{\star }$ of unit order, these large-$\unicode[STIX]{x1D70C}$ profiles have similar structure to the well-known logarithmic profiles for turbulent flow near a plane boundary (e.g. see §§ XIX.d and XX.c of Schlichting (Reference Schlichting1968)).

Hodographs of the horizontal velocity components graphed in figures 1, 2, 8 and 9 are presented in figure 11. Note that the strength of the meridional flow, illustrated by radial variation of the hodographs (the vertical in figure 11), is enhanced by boundary roughness and increases with decreasing $\unicode[STIX]{x1D70C}$.

Figure 11. Hodographs at four radial locations $\unicode[STIX]{x1D70C}=0.1$, 1.0, 10.0 and 100.0 (shown as dashed curves), together with those using model A and the laminar Bödewadt flow, from figure 3 (shown as solid curves). The dots on the curves denote unit values of $\unicode[STIX]{x1D701}$: $0,1,2,\ldots$.

7.1 Axial and supergradient flow

There has been some discussion in the literature whether it is correct to specify the circumferential speed at the top of the boundary layer in regions where the axial flow is exiting the layer (e.g. see Smith & Montgomery (Reference Smith and Montgomery2010) and references therein). Smith & Vogl (Reference Smith and Vogl2008) note that ‘It is probably incorrect to prescribe the tangential wind speed just above the boundary layer in the inner region, where the flow exits the boundary layer’. In a similar vein, Smith & Montgomery (Reference Smith and Montgomery2014) argue that prescription of the tangential wind at the top of the boundary layer where the flow is upwards ‘makes the physical problem ill-posed as the boundary layer itself should be allowed to determine the tangential momentum that it expels into the bulk vortex aloft’. These conjectures are based on a misunderstanding of the mathematical nature of boundary-layer flows. A viscous boundary layer is a dynamical structure that provides a smooth variation between the tangential velocity of fluid near a boundary and the velocity of the boundary itself. It is a general feature of these boundary layers that dynamical processes within the layer require an axial (perpendicular to the boundary) flow at the outer edge of the layer. That is, this axial flow is a dependent response to the imposition of an independent tangential velocity difference. In the present context, $v_{\infty }$ is the independently prescribed velocity difference and $w_{\infty }$ is the dependent response. This functional relationship holds whether $w_{\infty }$ is positive or negative. If the axial velocity were specified, then it is true that the circumferential velocity could not be. But this is not a physically reasonable approach. The boundary layer is forced by the circumferential velocity, with the axial velocity being determined by the internal dynamics of the boundary layer.

The classic laminar Bödewadt flow (see appendix C) and the turbulent boundary-layer flows presented in this article (see § 6) and in Part 2 all have positive axial flow (out of the boundary layer) for all radii; when the outer flow is rigid-body rotation, fluid is expelled from the boundary layer into the overlying fluid. Provided positive axial flow from the layer also occurs in the case of a swirling outer flow, such flow allows the boundary layer to communicate with – and indirectly to influence – this swirling flow. This communication is not very important to a tornado because there is little dynamical distinction between boundary-layer air and that above. However in a tropical cyclone, typically air within the boundary layer is warm and moist relative to the air above and so has the potential to drive convective motions. A mathematical consequence is that $v_{\infty }$ and $w_{\infty }$ are likely to be coupled functions of $r$ for a tropical cyclone, but not for a tornado.

Another issue mentioned in the literature is whether, in regions having vertical outflow from the boundary layer, supergradient flow is expelled and acts to enhance the vortical flow outside the layer (Smith & Vogl Reference Smith and Vogl2008; Kepert Reference Kepert2010a). The solutions presented above show that this is not the case. The supergradient flow is confined to the boundary layer and simply a manifestation of inertial oscillations. This suggests that models exhibiting supergradient flow at the top of the boundary layer may not be accurate.