3.1 The outer region: NH-QGEs

Within the fluid interior, inner $(i)$ and middle $(m)$ variables are identically zero. We introduce expansions in powers of ${\it\epsilon}$ of the form

(3.10) $$\begin{eqnarray}\displaystyle \boldsymbol{u}^{(o)}=\lim (\boldsymbol{u})^{o}=\boldsymbol{U}_{0}^{(o)}+{\it\epsilon}\boldsymbol{U}_{1}^{(o)}+{\it\epsilon}^{2}\boldsymbol{U}_{2}^{(o)}+\cdots \,, & & \displaystyle\end{eqnarray}$$

where $Ro\equiv {\it\epsilon}$ (Julien et al. Reference Julien, Knobloch, Milliff and Werne2006; Sprague et al. Reference Sprague, Julien, Knobloch and Werne2006; Julien & Knobloch Reference Julien and Knobloch2007). The leading-order mean component satisfies the motionless hydrostatic balance

(3.11a,b ) $$\begin{eqnarray}\displaystyle \overline{\boldsymbol{U}}_{0}^{(o)}=0,\quad \partial _{Z}\overline{P}_{0}^{(o)}=\frac{\widetilde{Ra}}{{\it\sigma}}\overline{{\it\Theta}}_{0}^{(o)}, & & \displaystyle\end{eqnarray}$$

together with $P_{0}^{\prime (o)}={\it\Theta}_{0}^{\prime (o)}=0$ . The leading-order convective dynamics is found to be incompressible and geostrophically balanced, i.e.

(3.12) $$\begin{eqnarray}\displaystyle \left.\begin{array}{@{}c@{}}\widehat{\boldsymbol{z}}\times \boldsymbol{U}_{0}^{\prime (o)}+\boldsymbol{{\rm\nabla}}P_{1}^{\prime (o)}=0,\\ \boldsymbol{{\rm\nabla}}\boldsymbol{\cdot }\boldsymbol{U}_{0}^{\prime (o)}=0,\end{array}\quad \text{or}\quad \mathscr{L}_{geo}\left(\begin{array}{@{}c@{}}\boldsymbol{U}_{0}^{\prime (o)}\\ P_{1}^{\prime (o)}\end{array}\right)=\mathbf{0},\right\} & & \displaystyle\end{eqnarray}$$

where $\mathscr{L}_{geo}$ denotes the geostrophic operator. By definition, all outer variables are independent of $z$ , and so $\boldsymbol{{\rm\nabla}}=\boldsymbol{{\rm\nabla}}_{\bot }\equiv (\partial _{x},\partial _{y},0)$ . If the small-scale $z$ dependence were retained, geostrophy would automatically imply the Proudman–Taylor (PT) constraint $\partial _{z}(\boldsymbol{U}_{0}^{\prime (o)},P_{1}^{\prime (o)})\equiv 0$ on the small vertical scale $z$ (Proudman Reference Proudman1916; Taylor Reference Taylor1923). The diagnostic balance given by (3.12) is solved by

(3.13a,b ) $$\begin{eqnarray}\boldsymbol{U}_{0}^{\prime (o)}=\boldsymbol{{\rm\nabla}}^{\bot }{\it\Psi}_{0}^{(o)}+W_{0}^{(o)}\widehat{\boldsymbol{z}},\quad P_{1}^{\prime (o)}={\it\Psi}_{0}^{(o)},\end{eqnarray}$$

for the streamfunction ${\it\Psi}_{0}^{(o)}(x,y,Z,t,{\it\tau})$ and vertical velocity $W_{0}^{(o)}(x,y,Z,t,{\it\tau})$ . Here, we adopt the definition $\boldsymbol{{\rm\nabla}}^{\bot }{\it\Psi}_{0}^{(o)}\equiv -\boldsymbol{{\rm\nabla}}_{\bot }\times {\it\Psi}_{0}^{(o)}\widehat{\boldsymbol{z}}$ with $\boldsymbol{{\rm\nabla}}^{\bot }=(-\partial _{y},\partial _{x},0)$ , so that the pressure is now identified as the geostrophic streamfunction. It follows that leading-order motions are horizontally non-divergent with $\boldsymbol{{\rm\nabla}}_{\bot }\boldsymbol{\cdot }\boldsymbol{U}_{0\bot }^{\prime (o)}=0$ . Three-dimensional incompressibility is captured at the next order, $\mathit{O}({\it\epsilon})$ , where

(3.14) $$\begin{eqnarray}\boldsymbol{{\rm\nabla}}_{\bot }\boldsymbol{\cdot }\boldsymbol{U}_{1\bot }^{\prime (o)}+\partial _{Z}W_{0}^{(o)}=0.\end{eqnarray}$$

This results in the production of subdominant ageostrophic motions $\boldsymbol{U}_{1\bot }^{\prime (o)}$ driven by vertical gradients in $W_{0}$ . The prognostic evolution of these variables is obtained from balances at next order,

(3.15) $$\begin{eqnarray}\displaystyle \mathscr{L}_{geo}\left(\begin{array}{@{}c@{}}\boldsymbol{U}_{1}^{\prime (o)}\\ P_{2}^{\prime (o)}\end{array}\right) & = & \displaystyle \text{RHS}\nonumber\\ \displaystyle & \equiv & \displaystyle \left(\begin{array}{@{}c@{}}-D_{0t}^{\bot }\boldsymbol{U}_{0}^{\prime (o)}-\partial _{Z}P_{1}\widehat{\boldsymbol{z}}+\displaystyle \frac{\widetilde{Ra}}{{\it\sigma}}{\it\Theta}_{1}^{\prime (o)}\widehat{\boldsymbol{z}}+{\rm\nabla}_{\bot }^{2}\boldsymbol{U}_{0}^{\prime (o)}\\ -\partial _{Z}W_{0}\end{array}\right),\end{eqnarray}$$

obtained by projecting RHS onto the null space of $\mathscr{L}_{geo}$ (Sprague et al. Reference Sprague, Julien, Knobloch and Werne2006; Calkins, Julien & Marti Reference Calkins, Julien and Marti2013). This amounts to performing the projections $\widehat{\boldsymbol{z}}\boldsymbol{\cdot }$ and $\boldsymbol{{\rm\nabla}}^{\bot }\boldsymbol{\cdot }$ on (3.15). On noting that $D_{0t}^{\bot }\equiv \partial _{t}+\boldsymbol{U}_{0\bot }^{\prime (o)}\boldsymbol{\cdot }\boldsymbol{{\rm\nabla}}_{\bot }$ , this procedure results in asymptotically reduced equations for the vertical vorticity ${\it\zeta}_{0}^{(o)}={\rm\nabla}_{\bot }^{2}{\it\Psi}_{0}^{(o)}$ , vertical velocity $W_{0}^{(o)}$ and thermal anomaly ${\it\Theta}_{1}^{\prime (o)}$ :

(3.16) $$\begin{eqnarray}\displaystyle & \displaystyle D_{0t}^{\bot }{\it\zeta}_{0}^{(o)}-\partial _{Z}W_{0}^{(o)}={\rm\nabla}_{\bot }^{2}{\it\zeta}_{0}^{(o)}, & \displaystyle\end{eqnarray}$$

(3.17) $$\begin{eqnarray}\displaystyle & \displaystyle D_{0t}^{\bot }W_{0}^{(o)}+\partial _{Z}{\it\Psi}_{0}^{(o)}=\frac{\widetilde{Ra}}{{\it\sigma}}{\it\Theta}_{1}^{\prime (o)}+{\rm\nabla}_{\bot }^{2}W_{0}^{(o)}, & \displaystyle\end{eqnarray}$$

(3.18) $$\begin{eqnarray}\displaystyle & \displaystyle D_{0t}^{\bot }{\it\Theta}_{1}^{\prime (o)}+W_{0}^{(o)}\partial _{Z}\overline{{\it\Theta}}_{0}^{(o)}=\frac{1}{{\it\sigma}}{\rm\nabla}_{\bot }^{2}{\it\Theta}_{1}^{\prime (o)}. & \displaystyle\end{eqnarray}$$

The evolution of the mean temperature field $\overline{{\it\Theta}}_{0}^{(o)}$ is deduced at $\mathit{O}({\it\epsilon}^{2})$ upon averaging over the fast scales $x,y,z,t$ :

(3.19) $$\begin{eqnarray}\displaystyle \partial _{{\it\tau}}\overline{{\it\Theta}}_{0}^{(o)}+\partial _{Z}\left(\overline{\overline{W_{0}^{(o)}{\it\Theta}_{1}^{\prime (o)}}}^{\mathscr{T}}\right) & = & \displaystyle \frac{1}{{\it\sigma}}\partial _{ZZ}\overline{{\it\Theta}}_{0}^{(o)}.\end{eqnarray}$$

In a statistically stationary state, it follows that

(3.20) $$\begin{eqnarray}Nu={\it\sigma}\left(\overline{\overline{W_{0}^{(o)}{\it\Theta}_{1}^{\prime (o)}}}^{\mathscr{T}}\right)-\partial _{Z}\overline{{\it\Theta}}_{0}^{(o)}.\end{eqnarray}$$

Equations (3.16)–(3.18), (3.11) and (3.19) constitute the asymptotically reduced system referred to as the NH-QGEs. A notable feature in the NH-QGEs is the absence of (higher-order) vertical advection. This is a hallmark characteristic of quasi-geostrophic theory. Equation (3.16) states that vertical vorticity, or toroidal motion, is affected by horizontal advection, vortex stretching arising from the linear Coriolis force and horizontal diffusion, while (3.17) shows that vertical motion is affected by horizontal advection, unbalanced pressure gradient, horizontal diffusion and buoyancy. The buoyancy forces are captured by the fluctuating and mean temperature equations (3.18) and (3.19).

The system (3.16)–(3.19) is accompanied by impenetrable boundary conditions

(3.21) $$\begin{eqnarray}W_{0}^{(o)}(0)=W_{0}^{(o)}(1)=0,\end{eqnarray}$$

together with thermal conditions, hereafter taken to be the fixed temperature conditions

(3.22a,b ) $$\begin{eqnarray}\overline{{\it\Theta}}_{0}^{(o)}(0)=1,\quad \overline{{\it\Theta}}_{0}^{(o)}(1)=0.\end{eqnarray}$$

In the presence of impenetrable boundaries, the boundary limits $Z\rightarrow 0$ or $Z\rightarrow 1$ of (3.18) for the temperature fluctuations ${\it\Theta}_{1}^{(o)}$ reduce to the advection–diffusion equation

(3.23) $$\begin{eqnarray}\displaystyle D_{0t}{\it\Theta}_{1}^{\prime (o)}=\frac{1}{{\it\sigma}}{\rm\nabla}_{\bot }^{2}{\it\Theta}_{1}^{\prime (o)}. & & \displaystyle\end{eqnarray}$$

The horizontally averaged variance $\overline{{\it\Theta}_{1}^{\prime (o)2}}$ for such an equation evolves according to

(3.24) $$\begin{eqnarray}\displaystyle \partial _{t}\overline{{\it\Theta}_{1}^{\prime (o)2}}=-\frac{1}{{\it\sigma}}\overline{|\boldsymbol{{\rm\nabla}}_{\bot }{\it\Theta}_{1}^{\prime (o)}|^{2}}. & & \displaystyle\end{eqnarray}$$

Therefore, $\overline{{\it\Theta}_{1}^{\prime (o)2}}$ decreases monotonically to zero in time, implying the implicit thermal boundary condition ${\it\Theta}_{1}^{\prime (o)}(0)={\it\Theta}_{1}^{\prime (o)}(1)=0$ . Together with the impenetrability condition (3.21), the vertical momentum equation (3.17) implies that motions along the horizontal boundaries are implicitly stress-free with

(3.25) $$\begin{eqnarray}\partial _{Z}{\it\Psi}_{0}^{(o)}(0)=\partial _{Z}{\it\Psi}_{0}^{(o)}(1)=0.\end{eqnarray}$$

If rapidly rotating RBC in the presence of stress-free boundary conditions is the primary objective, a well-posed closed system is obtained from (3.16)–(3.18), (3.11) and (3.19) together with impenetrable boundary conditions (3.21) and fixed mean temperature boundary conditions (3.22).

Figure 3. Comparison of the Nusselt number $Nu$ as a function of the reduced Rayleigh number $\widetilde{Ra}=RaE^{4/3}$ for NH-QGEs and DNS with (a) stress-free and (b) no-slip boundary conditions. The filled symbols illustrate DNS data obtained at $E=10^{-7}$ while the open symbols are from the reduced NH-QGEs. Data courtesy of Julien et al. (Reference Julien, Rubio, Grooms and Knobloch2012b ) and Stellmach et al. (Reference Stellmach, Lischper, Julien, Vasil, Cheng, Ribeiro, King and Aurnou2014).

3.1.1 Validity of the NH-QGEs

In the presence of stress-free boundary conditions, the NH-QGEs remain valid throughout the entire flow domain provided that geostrophy, equation (3.12), holds. This remains so provided that the local Rossby number $Ro_{l}\ll 1$ . Given $\boldsymbol{U}_{0}^{\ast }=\boldsymbol{U}_{0}^{(o)}{\it\nu}/L$ , one finds

(3.26) $$\begin{eqnarray}Ro_{l}=\frac{|\boldsymbol{U}_{0}^{\ast }|}{2{\it\Omega}L}=|\boldsymbol{U}_{0}^{(o)}|E\left(\frac{H}{L}\right)^{2}=|\boldsymbol{U}_{0}^{(o)}|E^{1/3}\Rightarrow |\boldsymbol{U}_{0}^{(o)}|\sim |{\it\zeta}_{0}^{(o)}|=o({\it\epsilon}^{-1}).\end{eqnarray}$$

(The Landau notation lower-case $o$ denotes a function that is of lower order of magnitude than a given function; that is, the function $o({\it\epsilon}^{-1})$ is of a lower order than the function ${\it\epsilon}^{-1}$ .) In a detailed investigation of the NH-QGEs, Julien et al. (Reference Julien, Knobloch, Rubio and Vasil2012a ) have shown that this criterion is violated at the transitional value

(3.27a,b ) $$\begin{eqnarray}\widetilde{Ra}_{tr}=\mathit{O}({\it\epsilon}^{-4/5}),\quad Ro_{tr}=\mathit{O}({\it\epsilon}^{3/5})\quad \text{as }{\it\epsilon}\rightarrow 0.\end{eqnarray}$$

In this regime, the thermal boundary layers experience a loss of geostrophic balance. This provides an upper bound for comparisons of the NH-QGEs with DNS with stress-free boundary conditions. Indeed, as illustrated in figure 3(a) for $E=10^{-7}$ , within the regime of validity, $\widetilde{Ra}\lesssim 70$ , good quantitative agreement in the heat transport measurements is observed (Stellmach et al. Reference Stellmach, Lischper, Julien, Vasil, Cheng, Ribeiro, King and Aurnou2014). Simulations (Julien et al. Reference Julien, Rubio, Grooms and Knobloch2012b ; Nieves et al. Reference Nieves, Rubio and Julien2014) of the NH-QGEs prior to this transition have revealed four different flow morphologies subsequently confirmed by both DNS (Stellmach et al. Reference Stellmach, Lischper, Julien, Vasil, Cheng, Ribeiro, King and Aurnou2014) and laboratory experiments (Cheng et al. Reference Cheng, Stellmach, Ribeiro, Grannan, King and Aurnou2015) as $\widetilde{Ra}$ is increased (see figure 4): a cellular regime, a convective Taylor column (CTC) regime consisting of weakly interacting shielded columns, a plume regime where the CTCs have lost stability and finally a geostrophic turbulence regime that is also associated with an inverse energy cascade that produces a depth-independent large-scale dipole vortex pair. All regimes are identifiable by changes in the heat transport exponent: the CTC regime exhibits a steep heat transport scaling law where $Nu\propto \widetilde{Ra}^{2.1}$ , whereas the geostrophic turbulence regime is characterized by a dissipation-free scaling law $Nu\propto {\it\sigma}^{-1/2}\widetilde{Ra}^{3/2}$ (Julien et al. Reference Julien, Knobloch, Rubio and Vasil2012a ) and an inverse turbulent energy cascade (Julien et al. Reference Julien, Knobloch, Rubio and Vasil2012a ,Reference Julien, Rubio, Grooms and Knobloch b ; Favier, Silvers & Proctor Reference Favier, Silvers and Proctor2014; Guervilly, Hughes & Jones Reference Guervilly, Hughes and Jones2014; Rubio et al. Reference Rubio, Julien, Knobloch and Weiss2014; Stellmach et al. Reference Stellmach, Lischper, Julien, Vasil, Cheng, Ribeiro, King and Aurnou2014). A rigorous upper-bound heat transport result for the NH-QGEs, $Nu<C\widetilde{Ra}^{3}$ , where $C$ is a constant, has also been reported (Grooms Reference Grooms2015; Grooms & Whitehead Reference Grooms and Whitehead2015).

Figure 4. Comparisons between laboratory experiments, DNS and NH-QGEs of flow morphologies of rotationally constrained Rayleigh–Bénard convection. As $\widetilde{Ra}$ increases, the flow transitions between cellular, convective Taylor columns, plumes and geostrophic turbulence regimes.

On the other hand, comparison between the NH-QGEs and DNS with no-slip boundaries and laboratory experiments (Stellmach et al. Reference Stellmach, Lischper, Julien, Vasil, Cheng, Ribeiro, King and Aurnou2014) reveals substantial differences (figure 3 b). Specifically, a steep scaling law in the heat transport is observed in the DNS study. Moreover, DNS and laboratory results both suggest that the steep scaling continues as $E\rightarrow 0$ (figure 1). It is now evident from the implicitly enforced stress-free boundary condition (3.25) that the reduced NH-QGE system and its solutions cannot be uniformly continued to impenetrable no-slip boundaries, where

(3.28) $$\begin{eqnarray}\boldsymbol{u}_{0}(0)=\boldsymbol{u}_{0}(1)=\mathbf{0}.\end{eqnarray}$$

In the presence of no-slip boundaries, it is well known that the viscous boundary layers are Ekman layers of depth $O(E^{1/2}H)$ (Greenspan Reference Greenspan1969). Within these layers, the geostrophic velocity field $\boldsymbol{U}_{0\bot }^{(o)}$ in the bulk must be reduced to zero. In the following, we proceed with an analysis of the rotationally constrained regime with the intent of extending the NH-QGEs to the case of no-slip boundaries.

3.2 Inner region: Ekman boundary layers

To avoid duplication, we focus on the lower Ekman boundary layer at $Z=0$ with non-dimensional depth $O({\it\epsilon}^{1/2})$ (an identical analysis applies for the upper boundary layer at $Z=1$ ). This depth arises as a result of the spatially anisotropic structure of rapidly rotating convection (Heard & Veronis Reference Heard and Veronis1971). In dimensional terms, the Ekman layer depth ${\it\epsilon}^{1/2}L\equiv E^{1/2}H$ since $L/H={\it\epsilon}=E^{1/3}$ .

The equations that capture the Ekman layer dynamics are obtained by taking the inner limit of the governing equations (2.1)–(2.3) about $Z=0$ using (3.9). We pose an asymptotic inner expansion in powers of ${\it\epsilon}^{1/2}$ of the form

(3.29) $$\begin{eqnarray}\displaystyle \boldsymbol{u}^{(i)}=\lim (\boldsymbol{u})^{i}=\boldsymbol{U}_{0}^{(i)}+{\it\epsilon}^{1/2}\boldsymbol{U}_{1/2}^{(i)}+{\it\epsilon}\boldsymbol{U}_{1}^{(i)}+\cdots \,, & & \displaystyle\end{eqnarray}$$

and utilize, a posteriori, knowledge that the middle layer variables have the asymptotic form

(3.30a,b ) $$\begin{eqnarray}\displaystyle \boldsymbol{u}^{(m)}={\it\epsilon}\boldsymbol{U}_{1}^{(m)}+\cdots \,,\quad p^{(m)}={\it\epsilon}^{2}P_{2}^{(m)}+\cdots \,, & & \displaystyle\end{eqnarray}$$

and therefore do not contribute to leading order. Subtraction of the contributions of the outer region then yields

(3.31) $$\begin{eqnarray}\displaystyle & \displaystyle \widehat{\boldsymbol{z}}\times \boldsymbol{U}_{0\bot }^{(i)}=-\boldsymbol{{\rm\nabla}}_{\bot }P_{1}^{(i)}+\partial _{{\it\mu}{\it\mu}}\boldsymbol{U}_{0\bot }^{(i)}, & \displaystyle\end{eqnarray}$$

(3.32) $$\begin{eqnarray}\displaystyle & \displaystyle 0=-\partial _{{\it\mu}}P_{1}^{(i)}, & \displaystyle\end{eqnarray}$$

(3.33) $$\begin{eqnarray}\displaystyle & \displaystyle \boldsymbol{{\rm\nabla}}_{\bot }\boldsymbol{\cdot }\boldsymbol{U}_{0\bot }^{(i)}+\partial _{{\it\mu}}W_{1/2}^{(i)}=0. & \displaystyle\end{eqnarray}$$

Given that the mean components are identically zero, the primed notation is omitted. It follows from (3.32) that the pressure within the Ekman boundary layer is the same as that outside it, and we thus take $P_{1}^{(i)}\equiv 0$ . The classical linear Ekman equations are therefore

(3.34) $$\begin{eqnarray}\displaystyle & \displaystyle \widehat{\boldsymbol{z}}\times \boldsymbol{U}_{0\bot }^{(i)}=\partial _{{\it\mu}{\it\mu}}\boldsymbol{U}_{0\bot }^{(i)}, & \displaystyle\end{eqnarray}$$

(3.35) $$\begin{eqnarray}\displaystyle & \displaystyle \boldsymbol{{\rm\nabla}}_{\bot }\boldsymbol{\cdot }\boldsymbol{U}_{0\bot }^{(i)}+\partial _{{\it\mu}}W_{1/2}^{(i)}=0. & \displaystyle\end{eqnarray}$$

After a simple reformulation and the introduction of no-slip boundary conditions, we have

(3.36a-c ) $$\begin{eqnarray}\displaystyle \left(\partial _{{\it\mu}}^{4}+1\right)\boldsymbol{U}_{0\bot }^{(i)}=0,\quad \boldsymbol{U}_{0\bot }^{(i)}(0)+\boldsymbol{U}_{0\bot }^{(o)}(0)=0,\quad \boldsymbol{U}_{0\bot }^{(i)}({\it\mu}\rightarrow \infty )=0, & & \displaystyle\end{eqnarray}$$

where we have utilized in advance that the leading-order middle layer variables $(\boldsymbol{U}_{0}^{(m)},P_{1}^{(m)})$ $\equiv 0$ . Since the flow within the Ekman layer is horizontally divergent, (3.35) implies the presence of vertical motions with velocity $w={\it\epsilon}^{1/2}W_{1/2}^{(i)}$ .

Figure 5. Sample Ekman spiral profiles, i.e. projections of the vertical profiles of the horizontal velocities $u(z),v(z)$ onto the $u,v$ plane, at two fixed horizontal locations in the lower viscous layer of the DNS at $RaE^{4/3}=20$ , $E=10^{-7}$ , ${\it\sigma}=7$ . The dashed line illustrates the analytic solution $(U_{0},V_{0})$ obtained from (3.37), (3.38). Open squares correspond to different vertical locations in the DNS boundary layer, computed using a vertical Chebyshev discretization with $385$ grid points.

The classical solution (Greenspan Reference Greenspan1969) is found within the Ekman layer at $Z=0$ and is given by

(3.37) $$\begin{eqnarray}\displaystyle & \displaystyle U_{0}^{(i)}(x,y,{\it\mu},t)=-\text{e}^{-{\it\mu}/\sqrt{2}}\left(U_{0}^{(o)}(x,y,0,t)\cos \frac{{\it\mu}}{\sqrt{2}}+V_{0}^{(o)}(x,y,0,t)\sin \frac{{\it\mu}}{\sqrt{2}}\right),\qquad & \displaystyle\end{eqnarray}$$

(3.38) $$\begin{eqnarray}\displaystyle & \displaystyle V_{0}^{(i)}(x,y,{\it\mu},t)=-\text{e}^{-{\it\mu}/\sqrt{2}}\left(V_{0}^{(o)}(x,y,0,t)\cos \frac{{\it\mu}}{\sqrt{2}}-U_{0}^{(o)}(x,y,0,t)\sin \frac{{\it\mu}}{\sqrt{2}}\right),\qquad & \displaystyle\end{eqnarray}$$

(3.39) $$\begin{eqnarray}\displaystyle & \displaystyle P_{1}^{(i)}(x,y,{\it\mu},t)=0. & \displaystyle\end{eqnarray}$$

Figure 5 illustrates sample boundary layer profiles obtained from DNS. The projected horizontal velocities are in perfect agreement with (3.37) and (3.38). These structures are robust throughout the boundary layer, thus providing confirmation of the existence of a linear Ekman layer (Stellmach et al. Reference Stellmach, Lischper, Julien, Vasil, Cheng, Ribeiro, King and Aurnou2014). Application of mass conservation (3.35) yields the inner and outer vertical velocities

(3.40) $$\begin{eqnarray}\displaystyle w & = & \displaystyle W_{0}^{(o)}(x,y,0,t)+{\it\epsilon}^{1/2}W_{1/2}^{(i)}(x,y,{\it\mu},t)\nonumber\\ \displaystyle & = & \displaystyle {\it\epsilon}^{1/2}\frac{1}{\sqrt{2}}{\it\zeta}_{0}^{(o)}(x,y,0,t)-{\it\epsilon}^{1/2}\frac{1}{\sqrt{2}}{\it\zeta}_{0}^{(o)}(x,y,0,t)\text{e}^{-{\it\mu}/\sqrt{2}}\left[\cos \frac{{\it\mu}}{\sqrt{2}}+\sin \frac{{\it\mu}}{\sqrt{2}}\right].\qquad\end{eqnarray}$$

As ${\it\mu}\rightarrow \infty$ , we see that $\lim \left(W_{1/2}^{(i)}\right)^{o}=0$ and hence that

(3.41) $$\begin{eqnarray}W_{0}^{(o)}(x,y,0,t)={\it\epsilon}^{1/2}{\textstyle \frac{1}{\sqrt{2}}}{\it\zeta}_{0}^{(o)}(x,y,0,t).\end{eqnarray}$$

This relation, often referred to as the Ekman pumping boundary condition, constitutes an exact parameterization of the linear Ekman layer in rotating RBC and represents Ekman pumping when $W_{0}^{(o)}>0$ and Ekman suction when $W_{0}^{(o)}<0$ . Hereafter, we do not distinguish between these two cases and refer to this phenomenon generically as Ekman pumping. Use of the Ekman pumping boundary condition alleviates the need to resolve the velocity dynamics that occurs within the Ekman layer. A similar analysis of the Ekman layer at $Z=1$ gives

(3.42) $$\begin{eqnarray}W_{0}^{(o)}(x,y,1,t)=-{\it\epsilon}^{1/2}{\textstyle \frac{1}{\sqrt{2}}}{\it\zeta}_{0}^{(o)}(x,y,1,t).\end{eqnarray}$$

Equations (3.41)–(3.42) exemplify the following important distinction between the asymptotic procedure performed here and the linear analyses of Niiler & Bisshopp (Reference Niiler and Bisshopp1965) and Heard & Veronis (Reference Heard and Veronis1971) which implicitly assume that $W_{1/2}^{(o)}=\pm (1/\sqrt{2}){\it\zeta}_{0}^{(o)}$ as opposed to $W_{0}^{(o)}=\pm {\it\epsilon}^{1/2}(1/\sqrt{2}){\it\zeta}_{0}^{(o)}$ (parameterized boundary conditions were not uncovered in these articles). The latter approach yields a perturbative analysis that captures a range of $\widetilde{Ra}$ for which the effect of Ekman pumping remains asymptotically close to the stress-free problem. However, for a sufficiently large $\widetilde{Ra}$ , determined below in § 3.3.1, the asymptotic expansion breaks down and becomes non-uniform. For completeness, this result is summarized in appendix A. We find that the maintenance of asymptotic uniformity requires that the pumping be elevated to contribute to the leading-order vertical velocity, as pursued here. This approach enables a complete exploration of Ekman pumping for all values of $\widetilde{Ra}$ for which the NH-QGEs remain valid.

Several studies (Barcilon Reference Barcilon1965; Faller & Kaylor Reference Faller and Kaylor1966; Dudis & Davis Reference Dudis and Davis1971) have established that solutions to the classical linear Ekman layer are unstable to small $\mathit{O}(E^{1/2}H)$ horizontal scale disturbances that evolve on rapid time scales that are filtered from the reduced dynamics. This occurs when the boundary layer Reynolds number $R_{E}=|\boldsymbol{U}_{0\bot }^{(o)}|E^{1/6}$ ${\sim}55$ . Although this is within the realm of possibility for the rotationally constrained regime according to (3.26), to alter the pumping parameterization such an instability must also reach amplitudes comparable to the bulk vorticity. No evidence of this is presently seen in DNS.

The Ekman layer solutions (3.37)–(3.38) together with the application of the exact parameterizations (3.41)–(3.42) indicate that the outer velocity fields $(\boldsymbol{U}_{0\bot }^{(o)},W_{0}^{(o)})$ can be continued uniformly to the computational boundaries at $Z=0,1$ . Evidence for this two-layer structure is provided in figure 6, which illustrates the root mean square (r.m.s.) velocity profiles at various magnifications obtained from DNS at $\widetilde{Ra}=20,E=10^{-7},{\it\sigma}=7$ . Specifically, the figure demonstrates that the velocity structure is unaffected by the presence of the thermal boundary layer, defined in terms of the maxima of the r.m.s. of ${\it\theta}$ (purple line, plot (b)). These results suggest that Ekman layers in DNS of rotating RBC may be replaced by the parameterized pumping boundary conditions

(3.43a,b ) $$\begin{eqnarray}w(x,y,0,t)={\it\epsilon}^{1/2}{\textstyle \frac{1}{\sqrt{2}}}{\it\zeta}(x,y,0,t),\quad w(x,y,1,t)=-{\it\epsilon}^{1/2}{\textstyle \frac{1}{\sqrt{2}}}{\it\zeta}(x,y,1,t).\end{eqnarray}$$

Stellmach et al. (Reference Stellmach, Lischper, Julien, Vasil, Cheng, Ribeiro, King and Aurnou2014) have demonstrated the accuracy of this boundary layer parameterization via comparison of the heat transport obtained from DNS with no-slip boundaries and DNS where the Ekman pumping conditions are used. With all other details being identical in the two DNS studies, excellent quantitative agreement is reached. This result establishes that it is the presence of Ekman layers that is responsible for the strong differences in heat transport observed between stress-free and no-slip boundaries (figure 3).

Figure 6. Root mean square velocity profiles obtained via DNS with no-slip boundaries at $\widetilde{Ra}=20$ , $E=10^{-7}$ , ${\it\sigma}=7$ . The blue solid vertical line is the r.m.s. horizontal velocity $\sqrt{\overline{u_{\bot }^{2}}}$ and the black solid vertical line is the r.m.s. vertical velocity $\sqrt{\overline{w^{2}}}$ . Shaded regions denote the variance obtained from a time series. (a) Entire layer, (b) magnification of the $\mathit{O}(E^{1/3}H)$ thermal boundary layer scale and (c) magnification of the $\mathit{O}(E^{1/2}H)$ lower Ekman boundary layer. The Ekman boundary layer is delineated by the red horizontal dashed line, while the fluctuating thermal boundary layer (which has no visible effect on the velocity profiles) is delineated by the purple horizontal dashed line. The numerical grid points are marked by square and circular symbols in (c). The bulk velocity profile plotted in (a) transitions at the Ekman layer illustrated in (c).

The inner component of the pumping velocity $W_{1/2}^{(i)}$ defined in (3.40) gives rise to an inner temperature fluctuation ${\it\Theta}^{\prime (i)}$ satisfying

(3.44) $$\begin{eqnarray}W_{1/2}^{(i)}\partial _{Z}\overline{{\it\Theta}}_{0}^{(o)}(0)=\frac{1}{{\it\sigma}}\partial _{{\it\mu}{\it\mu}}{\it\Theta}_{5/2}^{\prime (i)}.\end{eqnarray}$$

Utilizing (3.40), the solution to this equation is given by

(3.45) $$\begin{eqnarray}{\it\Theta}_{5/2}^{\prime (i)}(x,y,{\it\mu},t)=-\frac{1}{\sqrt{2}}\text{e}^{-{\it\mu}/\sqrt{2}}\left[\cos \frac{{\it\mu}}{\sqrt{2}}-\sin \frac{{\it\mu}}{\sqrt{2}}\right]{\it\zeta}_{0}^{(o)}(0)\partial _{Z}\overline{{\it\Theta}}_{0}^{(o)}(0).\end{eqnarray}$$

The limiting values as ${\it\mu}\rightarrow 0$ and ${\it\mu}\rightarrow \infty$ are

(3.46a,b ) $$\begin{eqnarray}{\it\Theta}^{\prime (i)}(0)=-{\it\epsilon}^{5/2}{\textstyle \frac{1}{\sqrt{2}}}{\it\zeta}_{0}^{(o)}(0)\,\partial _{Z}\overline{{\it\Theta}}_{0}^{(o)}(0),\quad {\it\Theta}^{\prime (i)}\left({\it\mu}\rightarrow \infty \right)=0.\end{eqnarray}$$

It thus follows that the temperature fluctuations within the Ekman layer are of magnitude ${\it\Theta}^{\prime (i)}=\mathit{O}({\it\epsilon}^{5/2}{\it\zeta}_{0}^{(o)}(0)\,\partial _{Z}\overline{{\it\Theta}}_{0}^{(o)}(0))$ . This observation yields an estimate of the convective heat transport,

(3.47) $$\begin{eqnarray}{\it\epsilon}^{3}W_{1/2}^{(i)}(0){\it\Theta}_{5/2}^{\prime (i)}(0)\sim {\it\epsilon}^{3}({\it\zeta}_{0}^{(o)}(0))^{2}\,\partial _{Z}\overline{{\it\Theta}}_{0}^{(o)}(0),\end{eqnarray}$$

which is smaller in magnitude by a factor of $\mathit{O}({\it\epsilon}^{2})$ when compared with that occurring in the convective interior, namely ${\it\epsilon}(W_{0}^{(o)}{\it\Theta}_{1}^{\prime (o)})$ . We can thus conclude that the observed enhancement of heat flux as measured by $Nu$ (figure 3) cannot occur directly within the Ekman layer given the vorticity bound (3.26).

Figure 7. Root mean square temperature profiles obtained via DNS with no-slip boundaries at $\widetilde{Ra}=Ra\,E^{4/3}=20$ , $E=10^{-7}$ , ${\it\sigma}=7$ . The blue solid vertical line is the r.m.s. temperature $E^{-1/3}\sqrt{\overline{{\it\theta}^{\prime 2}}}$ and the black solid vertical line is the r.m.s. mean temperature $\overline{{\it\theta}}$ . Shaded regions denote the variance obtained from a time series. (a) Entire layer, (b) magnification of the $\mathit{O}(E^{1/3}H)$ thermal boundary layer scale and (c) magnification of the Ekman boundary layer. The fluctuating thermal boundary layer is delineated by the purple horizontal dashed line, while the Ekman boundary layer is delineated by the red horizontal dashed line. The numerical grid points are marked by square and circular symbols in (c). The thermal profile exhibits no visible boundary layer structure on the Ekman layer scale (c).

Inspection of the r.m.s. thermal profiles at increasing magnification obtained at $E=10^{-7}$ shows no visible boundary layer structure in the vicinity of the Ekman layer (cf. figure 7 b,c). The r.m.s. profiles reveal that the thermal boundary layer extends much farther into the interior than the Ekman layer observed in figure 6. Indeed, we recall that the mean temperature $\overline{{\it\Theta}}_{0}^{(o)}(Z,{\it\tau})$ is an outer variable independent of the fast spatial variables ${\it\mu}$ and $z$ (Sprague et al. Reference Sprague, Julien, Knobloch and Werne2006).

3.2.1 The significance of Ekman pumping

It now remains to determine the nature of the thermal response in the immediate vicinity of the Ekman layer. Focusing again on the lower boundary, this requires consideration of how the reduced outer equation for the temperature fluctuations (3.18) can be continued to the physical boundaries. On Taylor-expanding all fluid variables within the Ekman layer, the outer component of the pumping velocity $W_{0}^{(o)}(0)$ induces outer temperature fluctuations ${\it\Theta}_{1}^{\prime (o)}(0)$ that satisfy

(3.48) $$\begin{eqnarray}(\partial _{t}+\boldsymbol{U}_{0\bot }^{(o)}(0)\boldsymbol{\cdot }\boldsymbol{{\rm\nabla}}_{\bot }){\it\Theta}_{1}^{\prime (o)}(0)+\frac{{\it\epsilon}^{1/2}}{\sqrt{2}}{\it\zeta}_{0}^{(o)}(0)\partial _{Z}\overline{{\it\Theta}}_{0}^{(o)}(0)=\frac{1}{{\it\sigma}}{\rm\nabla}_{\bot }^{2}{\it\Theta}_{1}^{\prime (o)}(0).\end{eqnarray}$$

Here, we have set $W_{0}^{(o)}(0)=({\it\epsilon}^{1/2}/\sqrt{2}){\it\zeta}_{0}^{(o)}(0)$ .

In a statistically stationary state, averaging the equation for the thermal variance obtained from (3.48) gives the balance

(3.49) $$\begin{eqnarray}\frac{{\it\epsilon}^{1/2}}{\sqrt{2}}\overline{\overline{{\it\Theta}_{1}^{\prime (o)}(0){\it\zeta}_{0}^{(o)}(0)}}^{\mathscr{T}}\partial _{Z}\overline{{\it\Theta}}_{0}^{(o)}(0)=-\frac{1}{{\it\sigma}}\overline{\overline{|\boldsymbol{{\rm\nabla}}_{\bot }{\it\Theta}_{1}^{\prime (o)}(0)|^{2}}}^{\mathscr{T}}.\end{eqnarray}$$

Clearly, in contrast to stress-free boundaries where ${\it\Theta}_{1}^{\prime (o)}(0)\equiv 0$ , pumping induces the enhanced thermal response

(3.50) $$\begin{eqnarray}{\it\Theta}_{1}^{\prime (o)}(0)=\mathit{O}({\it\epsilon}^{1/2}{\it\zeta}_{0}^{(o)}(0)\partial _{Z}\overline{{\it\Theta}}_{0}^{(o)}(0))\end{eqnarray}$$

immediately outside the Ekman layer. The associated enhancement in the convective flux is estimated as

(3.51) $$\begin{eqnarray}\overline{W_{0}^{(o)}(0){\it\Theta}_{1}^{\prime (o)}(0)}=\mathit{O}({\it\epsilon}({\it\zeta}_{0}^{(o)}(0))^{2}\partial _{Z}\overline{{\it\Theta}}_{0}^{(o)}(0)).\end{eqnarray}$$

This equation shows that the convective flux induced by Ekman pumping becomes as important as the conductive transport $-\partial _{Z}\overline{{\it\Theta}}_{0}^{(o)}(0)$ when

(3.52) $$\begin{eqnarray}{\it\zeta}_{0}^{(o)}(0)=\mathit{O}({\it\epsilon}^{-1/2}).\end{eqnarray}$$

This threshold is always achieved within the regime of asymptotic validity of the NH-QGEs that demands ${\it\zeta}_{0}^{(o)}(0)=o({\it\epsilon}^{-1})$ (see § 3.1.1). This result also indicates that the differences between stress-free and no-slip boundary conditions are asymptotically $\mathit{O}({\it\epsilon}^{1/2})$ -small for rotationally constrained RBC when ${\it\zeta}_{0}^{(o)}(0)=o\left({\it\epsilon}^{-1/2}\right)$ (see appendix A). Strong $\mathit{O}(1)$ departures are predicted to occur in the range

(3.53) $$\begin{eqnarray}\mathit{O}({\it\epsilon}^{-1/2})\leqslant {\it\zeta}_{0}^{(o)}(0)<\mathit{O}({\it\epsilon}^{-1}).\end{eqnarray}$$

As we show below, the vast majority of laboratory experiments and DNS studies fall within this range.

3.3 The middle region: thermal wind layer

It is evident from (3.50) that ${\it\Theta}_{1}^{\prime (o)}(0)$ cannot satisfy the thermal boundary condition ${\it\Theta}^{\prime (o)}(0)=0$ , implying that the thermal response to Ekman pumping in the NH-QGEs requires a boundary layer regularization. However, temperature fluctuations ${\it\Theta}^{\prime (i)}(0)=\mathit{O}({\it\epsilon}^{5/2}{\it\Theta}_{1}^{\prime (0)}(0))$ within the Ekman layer are too small to provide compensation (see (3.46a,b )). This indicates the existence of a middle $(m)$ region $\mathit{O}({\it\epsilon})$ in depth (Heard & Veronis Reference Heard and Veronis1971).

Within the middle region, the inner $(i)$ variables are identically zero and the bulk variables achieve their boundary values. We introduce an expansion of the form

(3.54) $$\begin{eqnarray}\displaystyle \boldsymbol{u}^{(m)}=\lim (\boldsymbol{u})^{m}=\boldsymbol{U}_{0}^{(m)}+{\it\epsilon}\boldsymbol{U}_{1}^{(m)}+{\it\epsilon}^{2}\boldsymbol{U}_{2}^{(m)}+\cdots \,, & & \displaystyle\end{eqnarray}$$

and retain geostrophic balance as the leading-order fluctuation balance in the middle region, as in (3.12). Thus, the Proudman–Taylor constraint $\partial _{z}(\boldsymbol{U}_{0}^{(m)},P_{1}^{(m)})=0$ implies $(\boldsymbol{U}_{0}^{(m)},P_{1}^{(m)})=0$ . Departures from the NH-QGEs (3.16)–(3.18), (3.11) and (3.19) due to Ekman pumping can be deduced from the following prognostic equations:

(3.55) $$\begin{eqnarray}\displaystyle & \displaystyle D_{0t}^{\bot }\boldsymbol{U}_{0\bot }^{\prime (o)}+\widehat{\boldsymbol{z}}\times \boldsymbol{U}_{1}=-\boldsymbol{{\rm\nabla}}_{\bot }P_{2}^{\prime }+{\rm\nabla}_{\bot }^{2}\boldsymbol{U}_{0\bot }^{\prime (o)}, & \displaystyle\end{eqnarray}$$

(3.56) $$\begin{eqnarray}\displaystyle & \displaystyle D_{0t}^{\bot }W_{0}^{\prime (o)}=-\partial _{z}P_{2}^{(m)}-\partial _{Z}P_{1}^{(o)}+\frac{\widetilde{Ra}}{{\it\sigma}}{\it\Theta}_{1}+{\rm\nabla}_{\bot }^{2}W_{0}^{\prime (o)}, & \displaystyle\end{eqnarray}$$

(3.57) $$\begin{eqnarray}\displaystyle & \displaystyle D_{0t}^{\bot }{\it\Theta}_{1}+W_{0}^{\prime (o)}(\partial _{z}{\it\Theta}_{1}^{(m)}+\partial _{Z}\overline{{\it\Theta}}_{0}^{(o)})=\frac{1}{{\it\sigma}}{\rm\nabla}^{2}{\it\Theta}_{1}, & \displaystyle\end{eqnarray}$$

(3.58) $$\begin{eqnarray}\displaystyle & \displaystyle \boldsymbol{{\rm\nabla}}\boldsymbol{\cdot }\boldsymbol{U}_{1}+\partial _{Z}W_{0}^{\prime (o)}=0. & \displaystyle\end{eqnarray}$$

Variables without superscripts contribute to both the outer $(o)$ and middle $(m)$ regions and can be separated by taking the limits (3.7)–(3.9). It should be noted that (3.57) contains the three-dimensional (3D) Laplacian ${\rm\nabla}^{2}\equiv \partial _{x}^{2}+\partial _{y}^{2}+\partial _{z}^{2}$ .

Inspection of the momentum equation (3.55) and the continuity equation (3.58) in the middle region yields the hydrostatic thermal wind balance

(3.59) $$\begin{eqnarray}\displaystyle & \displaystyle \widehat{\boldsymbol{z}}\times \boldsymbol{U}_{1}^{\prime (m)}=-\boldsymbol{{\rm\nabla}}_{\bot }P_{2}^{\prime (m)}, & \displaystyle\end{eqnarray}$$

(3.60) $$\begin{eqnarray}\displaystyle & \displaystyle \partial _{z}P_{2}^{\prime (m)}=\frac{\widetilde{Ra}}{{\it\sigma}}{\it\Theta}_{1}^{\prime (m)}, & \displaystyle\end{eqnarray}$$

(3.61a,b ) $$\begin{eqnarray}\boldsymbol{{\rm\nabla}}_{\bot }\boldsymbol{\cdot }\boldsymbol{U}_{1}^{\prime (m)}=0\quad \text{and}\quad W_{1}^{\prime (m)}\equiv 0,\end{eqnarray}$$

from which we find

(3.62a,b ) $$\begin{eqnarray}\partial _{z}\boldsymbol{U}_{1\bot }^{\prime (m)}=\frac{\widetilde{Ra}}{{\it\sigma}}\boldsymbol{{\rm\nabla}}^{\bot }{\it\Theta}_{1}^{\prime (m)},\quad \text{s.t. }\boldsymbol{U}_{1\bot }^{\prime (m)}=\boldsymbol{{\rm\nabla}}^{\bot }{\it\Psi}_{1}^{(m)},\quad P_{2}^{\prime (m)}={\it\Psi}_{1}^{(m)}.\end{eqnarray}$$

The upper and lower middle layer convective dynamics is thus completely reduced to the determination of ${\it\Theta}_{1}^{\prime (m)}$ , which from (3.57) evolves according to

(3.63) $$\begin{eqnarray}\displaystyle D_{0t}^{\bot }{\it\Theta}_{1}^{\prime (m)}+W_{0}^{\prime (o)}\partial _{z}\left(\overline{{\it\Theta}}_{1}^{(m)}+{\it\Theta}_{1}^{\prime (m)}\right)-\partial _{z}\left(\overline{W_{0}^{\prime ^{(o)}}{\it\Theta}_{1}^{\prime ^{(m)}}}\right)=\frac{1}{{\it\sigma}}{\rm\nabla}^{2}{\it\Theta}_{1}^{\prime (m)}. & & \displaystyle\end{eqnarray}$$

This couples to the mean state via

(3.64) $$\begin{eqnarray}\displaystyle & \displaystyle \partial _{z}\overline{P}_{2}^{(m)}=\frac{\widetilde{Ra}}{{\it\sigma}}\overline{{\it\Theta}}_{1}^{(m)}, & \displaystyle\end{eqnarray}$$

(3.65) $$\begin{eqnarray}\displaystyle & \displaystyle \partial _{t}\overline{{\it\Theta}}_{1}^{(m)}+\partial _{z}\left(\overline{W_{0}^{\prime ^{(o)}}{\it\Theta}_{1}^{\prime ^{(m)}}}\right)=\frac{1}{{\it\sigma}}\partial _{zz}\overline{{\it\Theta}}_{1}^{(m)}. & \displaystyle\end{eqnarray}$$

Equations (3.63) and (3.65) yield the thermal variance relation

(3.66) $$\begin{eqnarray}\frac{1}{2}\partial _{t}\left\langle \left(\overline{{\it\Theta}}_{1}^{(m)}\right)^{2}+\overline{\left({\it\Theta}_{1}^{\prime (m)}\right)^{2}}\right\rangle =-\frac{1}{{\it\sigma}}\left\langle \left(\partial _{z}\overline{{\it\Theta}}_{1}^{(m)}\right)^{2}+\overline{\left(\partial _{z}{\it\Theta}_{1}^{\prime (m)}\right)^{2}}\right\rangle .\end{eqnarray}$$

It follows that non-zero values of $\overline{{\it\Theta}}_{1}^{(m)},{\it\Theta}_{1}^{\prime (m)}$ only exist in the middle region if they are sustained through the regularizing boundary conditions

(3.67a,b ) $$\begin{eqnarray}\displaystyle {\it\Theta}_{1}^{\prime (o)}+{\it\Theta}_{1}^{\prime (m)}=0,\quad {\it\Theta}_{1}^{\prime (m)}(z\rightarrow \infty )=0, & & \displaystyle\end{eqnarray}$$

(3.68a,b ) $$\begin{eqnarray}\displaystyle \overline{{\it\Theta}}_{1}^{(o)}+\overline{{\it\Theta}}_{1}^{(m)}=0,\quad \overline{{\it\Theta}}_{1}^{(m)}(z\rightarrow \infty )=0 & & \displaystyle\end{eqnarray}$$

at $Z=0$ or $1$ . Importantly, we conclude from (3.67), (3.68) that equations (3.63), (3.65) are fully coupled to the interior dynamics. Averaging (3.65) over $t$ , followed by integration with respect to $z$ , generates the heat transport relation

(3.69) $$\begin{eqnarray}{\it\sigma}\left(\overline{\overline{W_{0}^{\prime (o)}{\it\Theta}_{1}^{\prime (m)}}}^{\mathscr{T}}\right)-\partial _{z}\overline{\overline{{\it\Theta}}_{1}^{(m)}}^{\mathscr{T}}=0.\end{eqnarray}$$

By application of the outer limit to this equation, the constant of integration must be identically zero. This relation indicates that there is no net heat flux associated with the middle layer dynamics.

We now see that within the upper and lower middle regions relation (2.5) yields

(3.70) $$\begin{eqnarray}Nu={\it\sigma}\left(\overline{\overline{W_{0}^{\prime (o)}\left({\it\Theta}_{1}^{\prime (o)}+{\it\Theta}_{1}^{\prime (m)}\right)}}^{\mathscr{T}}\right)-\left(\partial _{Z}\overline{{\it\Theta}}_{0}^{(o)}+\partial _{z}\overline{\overline{{\it\Theta}}_{1}^{(m)}}^{\mathscr{T}}\right).\end{eqnarray}$$

Given (3.69), we find

(3.71) $$\begin{eqnarray}Nu={\it\sigma}\left(\overline{\overline{W_{0}^{\prime (o)}{\it\Theta}_{1}^{\prime (o)}}}^{\mathscr{T}}\right)-\partial _{Z}\overline{{\it\Theta}}_{0}^{(o)},\end{eqnarray}$$

valid at every vertical level. This result states that the heat transport within the fluid layer is determined entirely within the bulk. Moreover, (3.48), (3.50) and (3.51) now imply that any enhancement in heat transport is entirely due to buoyancy production in ${\it\Theta}_{1}^{\prime (o)}$ arising through the nonlinear advection of the mean temperature gradient ${\it\epsilon}^{1/2}{\it\zeta}_{0}^{\prime (o)}\partial _{Z}\overline{{\it\Theta}}_{0}^{(o)}/\sqrt{2}$ generated by Ekman pumping.

Figure 8. Convective flux ${\it\sigma}\overline{w{\it\theta}}$ (vertical dashed line) and mean temperature gradient $-\partial _{z}\overline{{\it\theta}}=Nu-{\it\sigma}\overline{w{\it\theta}}$ (vertical solid line) obtained for DNS with no-slip boundaries at $\widetilde{Ra}=20,E=10^{-7},{\it\sigma}=7$ and average $Nu=17$ . Shaded regions denote the variance obtained from a time series. (a) Entire layer, (b) magnification of the $\mathit{O}(E^{1/3}H)$ thermal boundary layer scale and (c) magnification of the Ekman boundary layer. The fluctuating thermal boundary layer is delineated by the purple horizontal dashed line, while the Ekman boundary layer is delineated by the red horizontal dashed line. The numerical grid points are marked by square and circular symbols. Equipartition is reached within the thermal wind layer (b).

Direct numerical simulation results indicate that equipartition between convective and conductive heat transport is achieved within the middle layer (see figure 8). For comparison, for stress-free boundary conditions, equipartition occurs at a vertical depth well outside that associated with the middle layer.

3.3.1 Estimation of the transition threshold, $\widetilde{Ra}_{thres}$

In the following, we determine the threshold Rayleigh number $\widetilde{Ra}_{thres}$ at which Ekman pumping gains significance according to the criterion ${\it\zeta}_{0}^{(o)}(0)\sim {\it\epsilon}^{-1/2}$ in (3.53). This is achieved by assessing the $\widetilde{Ra}$ dependence of the outer fluid variables prior to the transition threshold where stress-free and no-slip boundary conditions are presumed to be asymptotically indistinguishable at leading order. Direct numerical simulations at $E=10^{-7}$ (Stellmach et al. Reference Stellmach, Lischper, Julien, Vasil, Cheng, Ribeiro, King and Aurnou2014, figure 3) have established that Ekman pumping has a significant effect on the heat transport within the laminar CTC regime. We therefore make an a priori assumption that the transition occurs within this regime. Simulations of the NH-QGEs for rotating Rayleigh–Bénard convection (Julien et al. Reference Julien, Rubio, Grooms and Knobloch2012b ) have established that the dynamics within the CTC regime exhibits power law scalings with respect to $\widetilde{Ra}$ in both the bulk and the thermal boundary layer. Hence, we pose the following scaling relations:

(3.72) $$\begin{eqnarray}\displaystyle \left.\begin{array}{@{}c@{}}\displaystyle W_{0}^{(o)}=\widetilde{Ra}^{{\hat{w}}}\widehat{W}_{0}^{(o)},\quad {\it\Psi}_{0}^{(o)}=\widetilde{Ra}^{\hat{{\it\psi}}}\widehat{{\it\Psi}}_{0}^{(o)},\quad {\it\zeta}_{0}^{(o)}=\widetilde{Ra}^{\hat{{\it\zeta}}}\widehat{{\it\zeta}}_{0}^{(o)},\quad {\it\Theta}_{1}^{\prime (o)}=\widetilde{Ra}^{\hat{{\it\theta}}}\widehat{{\it\Theta}}_{1}^{(o)},\\ \displaystyle \partial _{Z}\overline{{\it\Theta}}_{0}^{(o)}=\widetilde{Ra}^{\hat{\text{d}t}}\widehat{\partial _{Z}\overline{{\it\Theta}}}_{0}^{(o)},\quad Nu=\widetilde{Ra}^{\hat{{\it\beta}}}\widehat{Nu}.\end{array}\right\} & & \displaystyle\end{eqnarray}$$

On noting that $\partial _{t},\boldsymbol{{\rm\nabla}}_{\bot },\partial _{Z}=\mathit{O}(1)$ in the core region and that the CTC structures are known to be axisymmetric to leading order (Grooms et al. Reference Grooms, Julien, Weiss and Knobloch2010), the following balances hold in the NH-QGEs:

(3.73) $$\begin{eqnarray}\displaystyle & \displaystyle \partial _{t}{\it\zeta}_{0}^{(o)}\sim -\partial _{Z}W_{0}^{(o)}\sim {\rm\nabla}_{\bot }^{2}{\it\zeta}_{0}^{(o)}, & \displaystyle\end{eqnarray}$$

(3.74) $$\begin{eqnarray}\displaystyle & \displaystyle \partial _{t}W_{0}^{(o)}\sim \partial _{Z}{\it\Psi}_{1}^{(o)}\sim \widetilde{Ra}{\it\Theta}_{1}^{\prime (o)}\sim {\rm\nabla}_{\bot }^{2}W_{0}^{(o)}, & \displaystyle\end{eqnarray}$$

(3.75) $$\begin{eqnarray}\displaystyle & \displaystyle \partial _{t}{\it\Theta}_{1}^{\prime (o)}\sim W_{0}^{(o)}\partial _{Z}\overline{{\it\Theta}}_{0}^{(o)}\sim \displaystyle \frac{1}{{\it\sigma}}{\rm\nabla}_{\bot }^{2}{\it\Theta}_{1}^{\prime (o)}, & \displaystyle\end{eqnarray}$$

(3.76) $$\begin{eqnarray}\displaystyle & \displaystyle {\it\sigma}\overline{\overline{W_{0}^{(o)}{\it\Theta}_{1}^{\prime (o)}}}^{\mathscr{T}}\sim Nu. & \displaystyle\end{eqnarray}$$

The algebraic equations satisfied by the exponents defined in (3.72) are given by

(3.77a-c ) $$\begin{eqnarray}{\hat{w}}=\hat{{\it\psi}}=\hat{{\it\zeta}}=1+\hat{{\it\theta}},\quad {\hat{w}}+\hat{\text{d}t}=\hat{{\it\theta}},\quad {\hat{w}}+\hat{{\it\theta}}=\hat{{\it\beta}}.\end{eqnarray}$$

On assuming that $\hat{{\it\beta}}$ is known empirically, we obtain

(3.78a-c ) $$\begin{eqnarray}{\hat{w}}=\hat{{\it\psi}}=\hat{{\it\zeta}}=\frac{\hat{{\it\beta}}+1}{2},\quad \hat{{\it\theta}}=\frac{\hat{{\it\beta}}-1}{2},\quad \hat{\text{d}t}=-1.\end{eqnarray}$$

This result indicates that as the amplitude of convection intensifies with increasing $\widetilde{Ra}$ , the mean bulk temperature gradient approaches an increasingly well-mixed interior according to $\widetilde{Ra}^{-1}$ . This is a well-established result of the CTC regime (Sprague et al. Reference Sprague, Julien, Knobloch and Werne2006; Julien et al. Reference Julien, Rubio, Grooms and Knobloch2012b ) that is confirmed by the reduced simulations (Julien et al. Reference Julien, Rubio, Grooms and Knobloch2012b ), which yield $Nu\sim \widetilde{Ra}^{2.1}$ together with the bulk scalings

(3.79a-d ) $$\begin{eqnarray}\displaystyle \partial _{Z}\overline{{\it\Theta}}_{0}^{(o)}\sim \widetilde{Ra}^{-0.96},\quad W_{0}^{(o)}=\widetilde{Ra}^{1.53},\quad {\it\zeta}_{0}^{(o)}\sim \widetilde{Ra}^{1.55},\quad {\it\Theta}_{1}^{\prime (o)}\sim \widetilde{Ra}^{0.62} & & \displaystyle\end{eqnarray}$$

evaluated at $Z=1/2$ for all variables except vorticity which is evaluated at $Z=3/4$ owing to its antisymmetry. The empirically measured scalings are in good quantitative agreement with the choice $\hat{{\it\beta}}\approx 2$ , giving

(3.80a-c ) $$\begin{eqnarray}Nu\sim \widetilde{Ra}^{2},\quad W_{0}^{(o)}={\it\zeta}_{0}^{(o)}\sim \widetilde{Ra}^{3/2},\quad {\it\Theta}_{1}^{(o)}\sim \widetilde{Ra}^{1/2}.\end{eqnarray}$$

In the thermal boundary layers, where it is once again assumed that $\partial _{t},\boldsymbol{{\rm\nabla}}_{\bot }=\mathit{O}(1)$ , but now $\partial _{Z}=\widetilde{Ra}^{\hat{{\it\eta}}}\gg 1$ , the following balances hold:

(3.81) $$\begin{eqnarray}\displaystyle & \displaystyle \partial _{t}{\it\zeta}_{0}^{(o)}\sim -\partial _{Z}W_{0}^{(o)}\sim {\rm\nabla}_{\bot }^{2}{\it\zeta}_{0}^{(o)}, & \displaystyle\end{eqnarray}$$

(3.82) $$\begin{eqnarray}\displaystyle & \displaystyle \partial _{Z}{\it\Psi}_{1}^{(o)}\sim \widetilde{Ra}{\it\Theta}_{1}^{\prime (o)}, & \displaystyle\end{eqnarray}$$

(3.83) $$\begin{eqnarray}\displaystyle & \displaystyle \partial _{t}{\it\Theta}_{1}^{\prime (o)}\sim W_{0}^{(o)}\partial _{Z}\overline{{\it\Theta}}_{0}^{(o)}\sim \displaystyle \frac{1}{{\it\sigma}}{\rm\nabla}_{\bot }^{2}{\it\Theta}_{1}^{\prime (o)}, & \displaystyle\end{eqnarray}$$

(3.84) $$\begin{eqnarray}\displaystyle & \displaystyle {\it\sigma}\overline{\overline{W_{0}^{(o)}{\it\Theta}_{1}^{\prime (o)}}}^{\mathscr{T}}\sim \partial _{Z}\overline{{\it\Theta}}_{0}^{(o)}\sim Nu. & \displaystyle\end{eqnarray}$$

The algebraic equations satisfied by the exponents defined in (3.72) now satisfy

(3.85a-d ) $$\begin{eqnarray}\hat{{\it\eta}}+{\hat{w}}=\hat{{\it\psi}}=\hat{{\it\zeta}},\quad \hat{{\it\eta}}+\hat{{\it\psi}}=1+\hat{{\it\theta}},\quad {\hat{w}}+\hat{\text{d}t}=\hat{{\it\theta}},\quad {\hat{w}}+\hat{{\it\theta}}=\hat{\text{d}t}=\hat{{\it\beta}},\end{eqnarray}$$

with the solution

(3.86a-c ) $$\begin{eqnarray}\hat{{\it\eta}}=\hat{{\it\psi}}=\hat{{\it\zeta}}=\frac{\hat{{\it\beta}}+1}{2},\quad {\hat{w}}=0,\quad \hat{{\it\theta}}=\hat{\text{d}t}=\hat{{\it\beta}}.\end{eqnarray}$$

From (3.78) and (3.86), it now follows that the vorticity satisfies

(3.87) $$\begin{eqnarray}{\it\zeta}_{0}^{(o)}\sim \widetilde{Ra}^{(\hat{{\it\beta}}+1)/2}\widehat{{\it\zeta}}_{0}^{(o)},\end{eqnarray}$$

and hence that (see (3.53))

(3.88) $$\begin{eqnarray}{\it\zeta}_{0}^{(o)}(0)\sim {\it\epsilon}^{-1/2}\Rightarrow \widetilde{Ra}_{thres}=\mathit{O}({\it\epsilon}^{-1/(\hat{{\it\beta}}+1)})=\mathit{O}(E^{-1/(3(\hat{{\it\beta}}+1))}).\end{eqnarray}$$

Specifically, for the empirically observed CTC value of $\hat{{\it\beta}}\approx 2$ ,

(3.89) $$\begin{eqnarray}\widetilde{Ra}_{thres}=\mathit{O}({\it\epsilon}^{-1/3})=\mathit{O}(E^{-1/9}).\end{eqnarray}$$

At $E=10^{-7}$ this gives a value $\widetilde{Ra}_{thres}\sim 6.0$ . This is of the same magnitude as the critical Rayleigh number $\widetilde{Ra}_{c}=8.05$ (Heard & Veronis Reference Heard and Veronis1971), indicating an immediate departure from the stress-free case. This is a conclusion borne out by the recent DNS (figure 3(b)). At lower values of $E$ , it is found that $\widetilde{Ra}_{thres}>\widetilde{Ra}_{c}$ . From the interior and boundary layer scaling exponents (3.78) and (3.86), we also estimate the magnitude of Ekman pumping normalized by the midplane velocity as

(3.90) $$\begin{eqnarray}S=\frac{E^{1/6}{\it\zeta}_{0}^{(o)}(0)/\sqrt{2}}{w_{0}^{(o)}(\frac{1}{2})}=\mathit{O}(E^{1/6}),\end{eqnarray}$$

thus confirming the empirical DNS results (see figure 5 of Stellmach et al. (Reference Stellmach, Lischper, Julien, Vasil, Cheng, Ribeiro, King and Aurnou2014)).