4.1 Shallow water equations with thickness relaxation

The linearised form of the unforced ( $\boldsymbol{F}=\mathbf{0}$ ) shallow water equations (1.1) with radiative relaxation of the thickness on an equatorial beta plane may be written as (Matsuno Reference Matsuno1966; Gill Reference Gill1982)

(4.1a ) $$\begin{eqnarray}\displaystyle & h_{t}^{\prime }+u_{x}^{\prime }+v_{y}^{\prime }=-\unicode[STIX]{x1D705}h^{\prime }, & \displaystyle\end{eqnarray}$$

(4.1b ) $$\begin{eqnarray}\displaystyle & u_{t}^{\prime }-{\textstyle \frac{1}{2}}yv^{\prime }+h_{x}^{\prime }=-\unicode[STIX]{x1D6FE}u^{\prime }, & \displaystyle\end{eqnarray}$$

(4.1c ) $$\begin{eqnarray}\displaystyle & v_{t}^{\prime }+{\textstyle \frac{1}{2}}yu^{\prime }+h_{y}^{\prime }=-\unicode[STIX]{x1D6FE}v^{\prime }. & \displaystyle\end{eqnarray}$$

$h_{0}$

$c=\sqrt{g^{\prime }h_{0}}$

$L_{eq}=\sqrt{c/2\unicode[STIX]{x1D6FD}}$

$h_{0}$

$33^{\circ }$

$y=5$

$\unicode[STIX]{x1D705}=T/\unicode[STIX]{x1D70F}_{\mathit{rad}}$

$\unicode[STIX]{x1D6FE}=T/\unicode[STIX]{x1D70F}_{\mathit{fric}}$

$T=L_{eq}/c$

$\unicode[STIX]{x1D705}=7.88\times 10^{-3}$

$\unicode[STIX]{x1D6FE}=T/\unicode[STIX]{x1D70F}_{\mathit{fric}}=3.42\times 10^{-4}$

Following standard practice (e.g. Matsuno Reference Matsuno1966; Gill Reference Gill1982) we seek waves that are harmonic in longitude and time of the form $h^{\prime }(x,y,t)=\text{Re}\{{\hat{h}}(y)\exp (\text{i}(kx-\unicode[STIX]{x1D714}t))\}$ etc. With no dissipation ( $\unicode[STIX]{x1D705}=\unicode[STIX]{x1D6FE}=0$ ) the three equations (4.1) may be combined into a single ordinary differential equation (ODE) for the meridional velocity $\hat{v}(y)$ ,

(4.2) $$\begin{eqnarray}\frac{\text{d}^{2}\hat{v}}{\text{d}y^{2}}=(Ay^{2}-B)\hat{v},\quad A=\frac{1}{4},B=\unicode[STIX]{x1D714}^{2}-k^{2}-\frac{k}{2\unicode[STIX]{x1D714}},\end{eqnarray}$$

the same equation that governs a quantum harmonic oscillator. The solutions that decay as $y\rightarrow \pm \infty$ may be written using the Hermite polynomials $H_{n}(\unicode[STIX]{x1D709})$ as

(4.3) $$\begin{eqnarray}\hat{v}=H_{n}(\unicode[STIX]{x1D709})\exp (-\unicode[STIX]{x1D709}^{2}/2),\quad \unicode[STIX]{x1D709}=yA^{1/4}.\end{eqnarray}$$

The dispersion relation $A^{-1/2}B=2n+1$ for $n=-1,0,1,2,\ldots$ gives a cubic equation for  $\unicode[STIX]{x1D714}$ ,

(4.4) $$\begin{eqnarray}\unicode[STIX]{x1D714}^{2}-k^{2}-\frac{k}{2\unicode[STIX]{x1D714}}=n+\frac{1}{2}.\end{eqnarray}$$

The Rossby wave branch of solutions is characterised by $n\geqslant 1$ and $0<-\unicode[STIX]{x1D714}/k\ll 1$ . The other two roots give inertia–gravity waves with $|\unicode[STIX]{x1D714}|\geqslant 1$ (the dimensionless inertial frequency). The cases $n=0$ and $n=-1$ give the Yanai and Kelvin waves respectively. Figure 6 shows these different branches of the dispersion relation, all of which represent trapped waves localised within a few equatorial deformation scales $L_{\mathit{eq}}$ of the equator. The inertia–gravity, Kelvin and Yanai wave branches all exceed the inertial frequency $|\unicode[STIX]{x1D714}|=1$ . The inertia–gravity and Kelvin waves thus do not exist in QG theory, while the frequency of the QG form of the Yanai wave approaches 0 as $k\rightarrow 0$ like a Rossby wave. The mechanism responsible for setting the direction of equatorial jet in the shallow water models is expected to lie within QG theory, so we only consider the Rossby waves in the remainder of this paper.

The friction and radiative relaxation terms in (4.1 a – c ) change the $A$ and $B$ coefficients to (Yamagata & Philander Reference Yamagata and Philander1985)

(4.5a,b ) $$\begin{eqnarray}A=\frac{1}{4}\left(\frac{\unicode[STIX]{x1D714}+\text{i}\unicode[STIX]{x1D705}}{\unicode[STIX]{x1D714}+\text{i}\unicode[STIX]{x1D6FE}}\right),\quad B=\unicode[STIX]{x1D714}^{2}-k^{2}-\frac{k}{2\left(\unicode[STIX]{x1D714}+\text{i}\unicode[STIX]{x1D6FE}\right)}+\text{i}\unicode[STIX]{x1D714}(\unicode[STIX]{x1D705}+\unicode[STIX]{x1D6FE})-\unicode[STIX]{x1D705}\unicode[STIX]{x1D6FE},\end{eqnarray}$$

so the dispersion relation becomes

(4.6) $$\begin{eqnarray}\unicode[STIX]{x1D714}^{2}-k^{2}-\frac{k}{2(\unicode[STIX]{x1D714}+\text{i}\unicode[STIX]{x1D6FE})}+\text{i}\unicode[STIX]{x1D714}(\unicode[STIX]{x1D705}+\unicode[STIX]{x1D6FE})-\unicode[STIX]{x1D705}\unicode[STIX]{x1D6FE}=\left(n+\frac{1}{2}\right)\left(\frac{\unicode[STIX]{x1D714}+\text{i}\unicode[STIX]{x1D705}}{\unicode[STIX]{x1D714}+\text{i}\unicode[STIX]{x1D6FE}}\right)^{1/2}\quad \text{for}~n=1,2,\ldots .\end{eqnarray}$$

Figure 6.

Dimensionless dispersion relation (4.4) for equatorially trapped waves in the standard shallow water equations on the equatorial beta plane with no dissipation.

Figure 7.

The (a) real and (b) imaginary parts of the dimensionless equatorial Rossby wave frequency $\unicode[STIX]{x1D714}$ for the standard shallow water equations with no dissipation (green solid lines), dissipation by relaxation of the thickness (red dash-dot lines) and dissipation by friction (blue dotted lines). The three lines appear coincident in (a).

Figure 7 shows the real and imaginary parts of the complex frequency $\unicode[STIX]{x1D714}$ for the first few equatorial Rossby waves ( $n=1,2,3$ ) for different values of $\unicode[STIX]{x1D6FE}$ and  $\unicode[STIX]{x1D705}$ . The dispersion relation (4.4) is invariant under the transformation $(k,\unicode[STIX]{x1D714})\mapsto (-k,-\unicode[STIX]{x1D714})$ , so we choose to take $k>0$ . The Rossby wave frequency $\text{Re}(\unicode[STIX]{x1D714})$ is now negative, and virtually unchanged by either weak friction ( $\unicode[STIX]{x1D705}=0.001$ , $\unicode[STIX]{x1D6FE}=0$ ) or weak radiative relaxation ( $\unicode[STIX]{x1D705}=0$ , $\unicode[STIX]{x1D6FE}=0.001$ ). The decay rate $-\text{Im}(\unicode[STIX]{x1D714})$ due to cooling ( $\unicode[STIX]{x1D705}=0.001$ , $\unicode[STIX]{x1D6FE}=0$ ) is largest at $k=0$ and tends to zero as $k\rightarrow \infty$ . By contrast, the decay due to friction ( $\unicode[STIX]{x1D6FE}=0.001$ , $\unicode[STIX]{x1D705}=0$ ) is monotonically increasing with $k$ and $-\text{Im}(\unicode[STIX]{x1D714})$ tends to $-\unicode[STIX]{x1D6FE}$ as $k\rightarrow \infty$ . Figure 7(b) also suggests a symmetry about the line $\text{Im}(\unicode[STIX]{x1D714})=-\unicode[STIX]{x1D716}/2$ between small radiative relaxation ( $\unicode[STIX]{x1D705}=\unicode[STIX]{x1D716}$ and $\unicode[STIX]{x1D6FE}=0$ ) and small friction ( $\unicode[STIX]{x1D705}=0$ and $\unicode[STIX]{x1D6FE}=\unicode[STIX]{x1D716}$ ). We expand the roots of (4.6) as $\unicode[STIX]{x1D714}=\unicode[STIX]{x1D714}_{0}+\unicode[STIX]{x1D716}\,\unicode[STIX]{x1D714}_{1,\mathit{rad}}+\unicode[STIX]{x1D716}^{2}\,\unicode[STIX]{x1D714}_{2,\mathit{rad}}+\cdots$ for radiative relaxation, and $\unicode[STIX]{x1D714}=\unicode[STIX]{x1D714}_{0}+\unicode[STIX]{x1D716}\,\unicode[STIX]{x1D714}_{1,\mathit{fric}}+\unicode[STIX]{x1D716}^{2}\,\unicode[STIX]{x1D714}_{2,\mathit{fric}}+\cdots$ for friction, where $\unicode[STIX]{x1D714}_{0}$ is the real root of (4.4). The $O(\unicode[STIX]{x1D716})$ corrections are both purely imaginary:

(4.7a ) $$\begin{eqnarray}\displaystyle & \unicode[STIX]{x1D714}_{1,\mathit{rad}}={\textstyle \frac{1}{2}}\text{i}\unicode[STIX]{x1D714}_{0}(-4\unicode[STIX]{x1D714}_{0}^{2}+2n+1)/(k+4\unicode[STIX]{x1D714}_{0}^{3}), & \displaystyle\end{eqnarray}$$

(4.7b ) $$\begin{eqnarray}\displaystyle & \unicode[STIX]{x1D714}_{1,\mathit{fric}}={\textstyle \frac{1}{2}}\text{i}\unicode[STIX]{x1D714}_{0}(-4\unicode[STIX]{x1D714}_{0}^{2}-2n-1-2k/\unicode[STIX]{x1D714}_{0})/(k+4\unicode[STIX]{x1D714}_{0}^{3}). & \displaystyle\end{eqnarray}$$

$\text{Im}(\unicode[STIX]{x1D714})=-\unicode[STIX]{x1D716}/2$

(4.8) $$\begin{eqnarray}\unicode[STIX]{x1D714}_{1,\mathit{rad}}+\frac{\text{i}}{2}=\frac{\text{i}}{2}\left(\frac{k+(2n+1)\unicode[STIX]{x1D714}_{0}}{k+4\unicode[STIX]{x1D714}_{0}^{3}}\right)=-\!\left(\unicode[STIX]{x1D714}_{1,\mathit{fric}}+\frac{\text{i}}{2}\right)\end{eqnarray}$$

between the two dispersion relations truncated at $O(\unicode[STIX]{x1D716})$ .

4.2 Thermal shallow water equations

Linearising the thermal shallow water equations (1.2 a – c ) about a rest state with uniform thickness $h_{0}$ and temperature $\unicode[STIX]{x1D6E9}_{0}$ , and applying the equivalent non-dimensionalisation to § 4.1 based on the velocity scale $\sqrt{h_{0}\unicode[STIX]{x1D6E9}_{0}}$ gives

(4.9a ) $$\begin{eqnarray}\displaystyle & h_{t}^{\prime }+u_{x}^{\prime }+v_{y}^{\prime }=0, & \displaystyle\end{eqnarray}$$

(4.9b ) $$\begin{eqnarray}\displaystyle & \unicode[STIX]{x1D6E9}_{t}^{\prime }=-\unicode[STIX]{x1D705}(h^{\prime }+\unicode[STIX]{x1D6E9}^{\prime }), & \displaystyle\end{eqnarray}$$

(4.9c ) $$\begin{eqnarray}\displaystyle & u_{t}^{\prime }-{\textstyle \frac{1}{2}}yv^{\prime }+h_{x}^{\prime }+{\textstyle \frac{1}{2}}\unicode[STIX]{x1D6E9}_{x}^{\prime }=-\unicode[STIX]{x1D6FE}u^{\prime }, & \displaystyle\end{eqnarray}$$

(4.9d ) $$\begin{eqnarray}\displaystyle & v_{t}^{\prime }+{\textstyle \frac{1}{2}}yu^{\prime }+h_{y}^{\prime }+{\textstyle \frac{1}{2}}\unicode[STIX]{x1D6E9}_{y}^{\prime }=-\unicode[STIX]{x1D6FE}v^{\prime }. & \displaystyle\end{eqnarray}$$

$\hat{v}(y)$

(4.10a,b ) $$\begin{eqnarray}A=\frac{1}{4}\frac{\unicode[STIX]{x1D714}(\unicode[STIX]{x1D714}+\text{i}\unicode[STIX]{x1D705})}{(\unicode[STIX]{x1D714}+\text{i}\unicode[STIX]{x1D6FE})(\unicode[STIX]{x1D714}+\text{i}\unicode[STIX]{x1D705}/2)},\quad B=\frac{\unicode[STIX]{x1D714}(\unicode[STIX]{x1D714}+\text{i}\unicode[STIX]{x1D6FE})(\unicode[STIX]{x1D714}+\text{i}\unicode[STIX]{x1D705})}{\unicode[STIX]{x1D714}+\text{i}\unicode[STIX]{x1D705}/2}-k^{2}-\frac{k}{2(\unicode[STIX]{x1D714}+\text{i}\unicode[STIX]{x1D6FE})},\end{eqnarray}$$

so the dispersion relation for equatorially trapped waves is

(4.11) $$\begin{eqnarray}\frac{\unicode[STIX]{x1D714}(\unicode[STIX]{x1D714}+\text{i}\unicode[STIX]{x1D6FE})(\unicode[STIX]{x1D714}+\text{i}\unicode[STIX]{x1D705})}{\unicode[STIX]{x1D714}+\text{i}\unicode[STIX]{x1D705}/2}-k^{2}-\frac{k}{2(\unicode[STIX]{x1D714}+\text{i}\unicode[STIX]{x1D6FE})}=\left(n+\frac{1}{2}\right)\left(\frac{\unicode[STIX]{x1D714}(\unicode[STIX]{x1D714}+\text{i}\unicode[STIX]{x1D705})}{(\unicode[STIX]{x1D714}+\text{i}\unicode[STIX]{x1D6FE})(\unicode[STIX]{x1D714}+\text{i}\unicode[STIX]{x1D705}/2)}\right)^{1/2}.\end{eqnarray}$$

Figure 8 shows the real and imaginary parts of the complex frequency $\unicode[STIX]{x1D714}$ for the first few equatorial Rossby waves with $n\in \{1,2,3\}$ , and three different sets of $\unicode[STIX]{x1D705}$ and $\unicode[STIX]{x1D6FE}$ values. As before, the real part of $\unicode[STIX]{x1D714}$ is virtually unchanged by weak dissipation ( $\unicode[STIX]{x1D705}=0.002$ or $\unicode[STIX]{x1D6FE}=0.001$ ) while $\text{Im}(\unicode[STIX]{x1D714})$ becomes negative. The plot of $\text{Im}(\unicode[STIX]{x1D714})$ in figure 8 closely resembles our earlier figure 7, with the same apparent symmetry, if we now take $\unicode[STIX]{x1D705}=2\unicode[STIX]{x1D6FE}$ . A similar perturbation analysis to before shows that including this factor of 2 indeed makes $\text{Im}(\unicode[STIX]{x1D714})$ symmetric about $-\unicode[STIX]{x1D716}/2$ between the cases of small radiative relaxation ( $\unicode[STIX]{x1D705}=2\unicode[STIX]{x1D716}$ and $\unicode[STIX]{x1D6FE}=0$ ) and small friction ( $\unicode[STIX]{x1D705}=0$ and $\unicode[STIX]{x1D6FE}=\unicode[STIX]{x1D716}$ ).

Figure 8.

The real and imaginary parts of the dimensionless equatorial Rossby wave frequency $\unicode[STIX]{x1D714}$ for the thermal shallow water equations with no dissipation (green solid lines), dissipation by relaxation of the temperature (red dash-dot lines) and dissipation by friction (blue dotted lines). The three lines appear coincident in (a).

5.1 Shallow water equations with thickness relaxation

We consider the shallow water model (1.1) with radiative relaxation of the thickness, and no forcing, on an equatorial beta plane, and apply the non-dimensionalisation from § 4.1 to obtain

(5.1a ) $$\begin{eqnarray}\displaystyle & h_{t}+(hu)_{x}+(hv)_{y}=-\unicode[STIX]{x1D705}(h-1), & \displaystyle\end{eqnarray}$$

(5.1b ) $$\begin{eqnarray}\displaystyle & (hu)_{t}+\left(hu^{2}+{\textstyle \frac{1}{2}}h^{2}\right)_{x}+(huv)_{y}-{\textstyle \frac{1}{2}}yhv=-\unicode[STIX]{x1D6FE}hu-\unicode[STIX]{x1D705}u(h-1), & \displaystyle\end{eqnarray}$$

(5.1c ) $$\begin{eqnarray}\displaystyle & (hv)_{t}+(huv)_{x}+\left(hv^{2}+{\textstyle \frac{1}{2}}h^{2}\right)_{y}+{\textstyle \frac{1}{2}}yhu=-\unicode[STIX]{x1D6FE}hv-\unicode[STIX]{x1D705}v(h-1). & \displaystyle\end{eqnarray}$$

$h_{0}$

Following standard practice, we decompose the velocity and other fields as $\boldsymbol{u}=\langle \boldsymbol{u}\rangle +\boldsymbol{u}^{\prime }$ into a zonal mean $\langle \boldsymbol{u}\rangle$ and a deviation $\boldsymbol{u}^{\prime }$ . The zonal mean of the dimensionless mass conservation equation may then be written as

(5.2) $$\begin{eqnarray}\langle h\rangle _{t}+\langle hv\rangle _{y}=-\unicode[STIX]{x1D705}(\langle h\rangle -1).\end{eqnarray}$$

Similarly, the zonal means of the two momentum equations are

(5.3) $$\begin{eqnarray}\langle hu\rangle _{t}+\langle huv\rangle _{y}-{\textstyle \frac{1}{2}}y\langle hv\rangle =-\unicode[STIX]{x1D6FE}\langle hu\rangle -\unicode[STIX]{x1D705}(\langle hu\rangle -\langle u\rangle ),\end{eqnarray}$$

and

(5.4) $$\begin{eqnarray}\langle hv\rangle _{t}+\langle hv^{2}+{\textstyle \frac{1}{2}}h^{2}\rangle _{y}+{\textstyle \frac{1}{2}}y\langle hu\rangle =-\unicode[STIX]{x1D6FE}\langle hv\rangle -\unicode[STIX]{x1D705}(\langle hv\rangle -\langle v\rangle ).\end{eqnarray}$$

We now consider the form of solutions to the system (5.1a – c ) for initial conditions corresponding to a Rossby wave of small amplitude $O(a)$ superimposed on a rest state with unit thickness. For these initial conditions, and under the subsequent evolution, the thickness and velocity components satisfy the scalings

(5.5a-c ) $$\begin{eqnarray}h=1+\underbrace{h^{\prime }}_{O(a)}+\underbrace{(\langle h\rangle -1)}_{O(a^{2})},\quad u=\underbrace{u^{\prime }}_{O(a)}+\underbrace{\langle u\rangle }_{O(a^{2})},\quad v=\underbrace{v^{\prime }}_{O(a)}+\underbrace{\langle v\rangle }_{O(a^{2})}.\end{eqnarray}$$

We define $\langle \unicode[STIX]{x1D702}\rangle ^{\ast }=\langle h\unicode[STIX]{x1D702}\rangle /\langle h\rangle$ to be the thickness-weighted zonal mean of a quantity $\unicode[STIX]{x1D702}$ , as in Thuburn & Lagneau (Reference Thuburn and Lagneau1999). Using $\langle h\rangle =1+O(a^{2})$ , we can simplify the above to obtain evolution equations correct to $O(a^{2})$ in the form

(5.6a ) $$\begin{eqnarray}\displaystyle & \langle h\rangle _{t}+\langle v\rangle _{y}^{\ast }+\unicode[STIX]{x1D705}(\langle h\rangle -1)=0, & \displaystyle\end{eqnarray}$$

(5.6b ) $$\begin{eqnarray}\displaystyle & \langle u\rangle _{t}^{\ast }-{\textstyle \frac{1}{2}}y\langle v\rangle ^{\ast }+\unicode[STIX]{x1D6FE}\langle u\rangle ^{\ast }=-\langle u^{\prime }v^{\prime }\rangle _{y}-\unicode[STIX]{x1D705}\langle h^{\prime }u^{\prime }\rangle , & \displaystyle\end{eqnarray}$$

(5.6c ) $$\begin{eqnarray}\displaystyle & \langle v\rangle _{t}^{\ast }+{\textstyle \frac{1}{2}}y\langle u\rangle ^{\ast }+\unicode[STIX]{x1D6FE}\langle v\rangle ^{\ast }+\langle h\rangle _{y}=-\langle v^{\prime }v^{\prime }\rangle _{y}-\unicode[STIX]{x1D705}\langle h^{\prime }v^{\prime }\rangle -{\textstyle \frac{1}{2}}\langle h^{\prime 2}\rangle _{y}. & \displaystyle\end{eqnarray}$$

$\langle h\rangle$

$\langle u\rangle ^{\ast }$

$\langle v\rangle ^{\ast }$

$h^{\prime }$

$u^{\prime }$

$v^{\prime }$

$\langle u^{\prime }v^{\prime }\rangle$

$\langle v^{\prime }v^{\prime }\rangle$

$-\langle h^{\prime 2}\rangle _{y}/2$

(5.7) $$\begin{eqnarray}h^{2}=1+2h^{\prime }+2(\langle h\rangle -1)+h^{\prime 2}+O(a^{3}).\end{eqnarray}$$

There are also radiative relaxation terms proportional to the so-called bolus velocity components $\langle h^{\prime }u^{\prime }\rangle$ and $\langle h^{\prime }v^{\prime }\rangle$ . These terms would be absent for the momentum-conserving variant of the shallow water equations with relaxation of the thickness.

The effect of the Rossby wave thus appears as an effective force on the right-hand sides of the evolution equations for $\langle u\rangle ^{\ast }$ and $\langle v\rangle ^{\ast }$ . Our use of the thickness-weighted means $\langle u\rangle ^{\ast }$ and $\langle v\rangle ^{\ast }$ in place of the unweighted means $\langle u\rangle$ and $\langle v\rangle$ eliminates a $\langle h^{\prime }v^{\prime }\rangle$ term that would otherwise appear in (5.6a ). This is similar to the transformed Eulerian mean (TEM) approach for stratified fluids (Andrews & McIntyre Reference Andrews and McIntyre1976a , Reference Andrews and McIntyre1978).

A further simplification is possible by assuming that geostrophic balance holds in the meridional momentum equation (5.6c ), since the meridional length scale and velocity component are both smaller than the zonal length scale and zonal velocity component. The same scaling argument is used in the semi-geostrophic theory of atmospheric front formation. The time derivative of this meridional geostrophic balance condition gives

(5.8) $$\begin{eqnarray}{\textstyle \frac{1}{2}}y\langle u\rangle _{t}^{\ast }+\langle h\rangle _{yt}=-{\textstyle \frac{1}{2}}\langle h^{\prime 2}\rangle _{yt},\end{eqnarray}$$

which can be used in place of (5.6c ) to obtain a closed system with (5.6a ) and (5.6b ).

We now evaluate these right-hand sides using the equatorially trapped Rossby wave solutions in § 4.1. These represent small perturbations about a state of rest with unit thickness in dimensionless variables. The meridional velocity perturbation is $v^{\prime }=\unicode[STIX]{x1D6FC}\,\text{Re}\{\hat{v}(y)\exp (\text{i}(kx-\unicode[STIX]{x1D714}t))\}$ , with $\hat{v}(y)$ defined by (4.3) and (4.5), and the constant $\unicode[STIX]{x1D6FC}$ sets the amplitude. The corresponding zonal velocity and thickness perturbations are $u^{\prime }=\unicode[STIX]{x1D6FC}\,\text{Re}\{\hat{u} (y)\exp (\text{i}(kx-\unicode[STIX]{x1D714}t))\}$ and $h^{\prime }=\unicode[STIX]{x1D6FC}\,\text{Re}\{{\hat{h}}(y)\exp (\text{i}(kx-\unicode[STIX]{x1D714}t))\}$ , where

(5.9a,b ) $$\begin{eqnarray}\hat{u} (y)=\text{i}\left(\frac{k(\text{d}\hat{v}/\text{d}y)-{\textstyle \frac{1}{2}}y(\unicode[STIX]{x1D714}+\text{i}\unicode[STIX]{x1D705})\hat{v}}{k^{2}-(\unicode[STIX]{x1D714}+\text{i}\unicode[STIX]{x1D6FE})(\unicode[STIX]{x1D714}+\text{i}\unicode[STIX]{x1D705})}\right),\quad {\hat{h}}(y)=-\text{i}\left(\frac{(\text{i}\unicode[STIX]{x1D714}-\unicode[STIX]{x1D6FE})\hat{u} +{\textstyle \frac{1}{2}}y\hat{v}}{k}\right),\end{eqnarray}$$

for $k\neq 0$ . The denominator in the expression for $\hat{u} (y)$ vanishes when $k^{2}=(\unicode[STIX]{x1D714}+\text{i}\unicode[STIX]{x1D6FE})(\unicode[STIX]{x1D714}+\text{i}\unicode[STIX]{x1D705})$ . This gives the dispersion relation for equatorially trapped Kelvin waves, which are distinguished by having zero meridional velocity ( $\hat{v}=0$ ), and frequencies $\unicode[STIX]{x1D714}=\pm k$ in the absence of dissipation.

To allow a fair comparison between the effective forces generated by waves with different $n$ and $k$ values, we normalise the disturbance amplitudes by choosing $\unicode[STIX]{x1D6FC}$ so that

(5.10) $$\begin{eqnarray}\frac{1}{2}\int \langle u^{\prime 2}\rangle +\langle v^{\prime 2}\rangle +\langle h^{\prime 2}\rangle \,\text{d}y=a^{2}.\end{eqnarray}$$

The left-hand side of (5.10) is the quadratic approximation to the disturbance pseudoenergy, an exact conserved quantity of the unforced, non-dissipative shallow water equations that takes the above form when expanded for small-amplitude disturbances from a rest state of constant thickness (Shepherd Reference Shepherd1993). Using a standard formula for the average of a product of sinusoidal disturbances represented by the real parts of complex exponentials, we can compute the integrand in (5.10), and the terms on the right-hand sides of (5.6b ) and (5.6c ), using

(5.11) $$\begin{eqnarray}\langle u^{\prime }v^{\prime }\rangle ={\textstyle \frac{1}{2}}\text{Re}(\hat{u} (y)\hat{v}(y)^{\dagger }),\end{eqnarray}$$

where the superscript $\text{}^{\dagger }$ denotes a complex conjugate, and similarly for the other quadratic combinations of $h^{\prime }$ , $u^{\prime }$ , $v^{\prime }$ .

Figure 9.

The effective zonal forcing terms $-\langle u^{\prime }v^{\prime }\rangle _{y}$ and $-\unicode[STIX]{x1D705}\langle h^{\prime }u^{\prime }\rangle$ on the right-hand side of (5.6b ), and their sum, for (a) dissipation solely by radiative relaxation ( $\unicode[STIX]{x1D705}=0.001,\unicode[STIX]{x1D6FE}=0$ ) and (b) dissipation solely by friction ( $\unicode[STIX]{x1D705}=0,\unicode[STIX]{x1D6FE}=0.001$ ). All terms are calculated for $n=1$ , $k=1$ , and $a=1$ for scaling. In (b) the sum equals the first term as $\unicode[STIX]{x1D705}=0$ . Here and in subsequent figures the legend applies to both plots.

Figure 10.

The effective zonal forcing terms $-\langle u^{\prime }v^{\prime }\rangle _{y}$ and $-\unicode[STIX]{x1D705}\langle h^{\prime }u^{\prime }\rangle$ on the right-hand side of (5.6b ), and their sum, for (a) dissipation solely by radiative relaxation ( $\unicode[STIX]{x1D705}=0.001,\unicode[STIX]{x1D6FE}=0$ ) and (b) dissipation solely by friction ( $\unicode[STIX]{x1D705}=0,\unicode[STIX]{x1D6FE}=0.001$ ). All terms are calculated for $n=2$ , $k=1$ , and $a=1$ for scaling. In (b) the sum equals the first term as $\unicode[STIX]{x1D705}=0$ .

Figures 9 and 10 show the two quantities $-\langle u^{\prime }v^{\prime }\rangle _{y}$ and $-\unicode[STIX]{x1D705}\langle h^{\prime }u^{\prime }\rangle$ on the right-hand side of (5.6b ), and their sum, for waves with $k=1$ and $n\in \{1,2\}$ for dissipation by either Rayleigh friction ( $\unicode[STIX]{x1D6FE}>0$ , $\unicode[STIX]{x1D705}=0$ ) or radiative relaxation ( $\unicode[STIX]{x1D6FE}=0$ , $\unicode[STIX]{x1D705}>0$ ). The plots show the quantities divided by $a^{2}$ , or as given by setting $a=1$ . The Reynolds stress $\langle u^{\prime }v^{\prime }\rangle$ vanishes for undamped waves, since $\unicode[STIX]{x1D714}$ , $A$ , and $\hat{v}$ are then all real, while $\hat{u}$ and ${\hat{h}}$ are purely imaginary. The bolus velocity term $-\unicode[STIX]{x1D705}\langle h^{\prime }u^{\prime }\rangle$ is explicitly proportional to $\unicode[STIX]{x1D705}$ , and so vanishes for $\unicode[STIX]{x1D705}=0$ .

A perturbative analysis similar to that in § 4.1 shows that the Reynolds stress $\langle u^{\prime }v^{\prime }\rangle$ is proportional to $\unicode[STIX]{x1D705}-\unicode[STIX]{x1D6FE}$ for small $\unicode[STIX]{x1D705}$ and $\unicode[STIX]{x1D6FE}$ . This explains the antisymmetry visible when comparing case (a) with $\unicode[STIX]{x1D705}>0$ , $\unicode[STIX]{x1D6FE}=0$ to case (b) with $\unicode[STIX]{x1D705}=0$ , $\unicode[STIX]{x1D6FE}>0$ in each of figures 9 and 10. Increasing $k$ while holding $n$ , $\unicode[STIX]{x1D705}$ , $\unicode[STIX]{x1D6FE}$ fixed increases the magnitude of $\langle u^{\prime }v^{\prime }\rangle$ without changing its general shape, as does increasing $\unicode[STIX]{x1D705}$ or $\unicode[STIX]{x1D6FE}$ while holding $k$ and $n$ fixed.

The Reynolds stress convergence is close to zero at the equator when $n$ is odd, leaving just the contribution from the bolus velocity visible in figure 9(a). However, the Reynolds stress convergence has a local extremum on the equator when $n$ is even. Its sign for $n=2$ is such as to produce eastward acceleration at the equator under radiative relaxation (figure 10 a), and an westward acceleration at the equator under Rayleigh friction (figure 10 b). We can infer the sign of the acceleration directly from the right-hand side of (5.6b ) on the equator. Since $y=0$ , the Coriolis contribution from the mean meridional velocity $\langle v\rangle ^{\ast }$ does not contribute to the left-hand side of (5.6b ). This leaves the ordinary differential equation

(5.12) $$\begin{eqnarray}\frac{\text{d}}{\text{d}t}\langle u\rangle ^{\ast }|_{y=0}+\unicode[STIX]{x1D6FE}\langle u\rangle ^{\ast }|_{y=0}=-\text{e}^{-\unicode[STIX]{x1D70E}t}(\langle u^{\prime }v^{\prime }\rangle _{y}+\unicode[STIX]{x1D705}\langle h^{\prime }u^{\prime }\rangle )|_{y=0,t=0},\end{eqnarray}$$

where $\unicode[STIX]{x1D70E}=-2\text{Im}\,\unicode[STIX]{x1D714}$ is twice the decay rate of the Rossby wave calculated in § 4.1. The quadratic products on the right-hand side of (5.6b ) thus decay in proportion to $\exp (-\unicode[STIX]{x1D70E}t)$ . The solution of (5.12) for $\unicode[STIX]{x1D6FE}\neq \unicode[STIX]{x1D70E}$ is

(5.13) $$\begin{eqnarray}\langle u\rangle ^{\ast }|_{y=0}=\left[\frac{\text{e}^{-\unicode[STIX]{x1D6FE}t}-\text{e}^{-\unicode[STIX]{x1D70E}t}}{\unicode[STIX]{x1D70E}-\unicode[STIX]{x1D6FE}}\right](\langle u^{\prime }v^{\prime }\rangle _{y}+\unicode[STIX]{x1D705}\langle h^{\prime }u^{\prime }\rangle )|_{y=0,t=0}.\end{eqnarray}$$

The time-dependent coefficient in square brackets $[\cdots ]$ is always positive for $t>0$ , so the direction of the zonal mean flow $\langle u\rangle ^{\ast }$ at the equator is set by the sign of the combined Reynolds stress and bolus velocity term evaluated at $y=0$ and $t=0$ . This is consistent with the directions of the equatorial jets found in numerical simulations for different rates of Rayleigh friction and radiative relaxation (Scott & Polvani Reference Scott and Polvani2007, Reference Scott and Polvani2008; Warneford & Dellar Reference Warneford and Dellar2014).

However, away from the equator one would need to compute the solution to the full response problem coupling $\langle h\rangle$ , $\langle u\rangle ^{\ast }$ , $\langle v\rangle ^{\ast }$ as three functions of $y$ and $t$ to find the mean flow produced by the wave sources on the right-hand sides of (5.6b ) and (5.6c ). Figure 11 shows the three terms $-\langle h^{\prime 2}\rangle _{y}/2$ , $-\langle v^{\prime }v^{\prime }\rangle _{y}$ and $-\unicode[STIX]{x1D705}\langle h^{\prime }u^{\prime }\rangle$ on the right-hand side of (5.6c ), and their sum, for $k=1$ , $n\in \{1,2\}$ and dissipation solely by radiative relaxation ( $\unicode[STIX]{x1D705}=0.001,\unicode[STIX]{x1D6FE}=0$ ). The first two terms are non-zero even with no dissipation, and typically much larger in magnitude than either the bolus terms or the Reynolds stress convergence in the zonal equation.

Omitting the $\unicode[STIX]{x1D705}$ terms from the right-hand sides of (5.1b ) and (5.1c ) leads to a shallow water model that relaxes the thickness field but conserves momentum. This model is no less justified than the model that supposes that $\unicode[STIX]{x1D705}$ should not appear in (1.1b ) for evolving the velocity $\boldsymbol{u}$ . The contributions $\langle h^{\prime }u^{\prime }\rangle$ and $\langle h^{\prime }v^{\prime }\rangle$ from the bolus velocity would then be absent in the right-hand sides of the analogues of (5.6b ) and (5.6c ) for this model. Figures 9(a) and 10(a) show that these contributions are in any case small compared with the contribution from the Reynolds stress convergence. This is consistent with the directions of the equatorial jets in the two models, with and without momentum conservation, showing the same behaviour in simulations (Warneford & Dellar Reference Warneford and Dellar2014).

Figure 11.

The effective meridional forcing terms $-\langle h^{\prime 2}\rangle _{y}/2$ , $-\langle v^{\prime }v^{\prime }\rangle _{y}$ and $-\unicode[STIX]{x1D705}\langle h^{\prime }u^{\prime }\rangle$ on the right-hand side of (5.6c ), and their sum, for dissipation solely by radiative relaxation ( $\unicode[STIX]{x1D705}=0.001,\unicode[STIX]{x1D6FE}=0$ ). All terms are calculated for $k=1$ , $a=1$ for scaling, and either (a)  $n=1$ or (b)  $n=2$ . The bolus velocity term $-\unicode[STIX]{x1D705}\langle h^{\prime }u^{\prime }\rangle$ is much smaller than the other two terms, which are not proportional to  $\unicode[STIX]{x1D705}$ .

5.2 Thermal shallow water equations

We now apply the same approach to the dimensionless form of the unforced thermal shallow water equations (1.2 a –c) on an equatorial beta plane. The mass and zonal momentum equations are

(5.14) $$\begin{eqnarray}\langle h\rangle _{t}+\langle v\rangle _{y}^{\ast }=0,\end{eqnarray}$$

and

(5.15) $$\begin{eqnarray}\langle u\rangle _{t}^{\ast }-{\textstyle \frac{1}{2}}y\langle v\rangle ^{\ast }+\unicode[STIX]{x1D6FE}\langle u\rangle ^{\ast }=-\langle u^{\prime }v^{\prime }\rangle _{y}.\end{eqnarray}$$

There are no $\unicode[STIX]{x1D705}$ terms in these equations, as radiative relaxation conserves mass and momentum in our thermal shallow water model. The temperature $\unicode[STIX]{x1D6E9}$ also does not appear in these equations, because the zonal pressure gradient disappears under zonal averaging.

To calculate the thermal contributions to the other equations, it is convenient to decompose the temperature as

(5.16) $$\begin{eqnarray}\unicode[STIX]{x1D6E9}=1+\unicode[STIX]{x1D703}=1+\underbrace{\unicode[STIX]{x1D703}^{\prime }}_{O(a)}+\underbrace{\langle \unicode[STIX]{x1D703}\rangle }_{O(a^{2})}.\end{eqnarray}$$

This decomposition eliminates $O(1)$ terms proportional to the reference temperature $\unicode[STIX]{x1D6E9}_{0}$ , which is scaled to unity by the non-dimensionalisation in § 4.2. Although $\unicode[STIX]{x1D703}^{\prime }=\unicode[STIX]{x1D6E9}^{\prime }$ , we use $\unicode[STIX]{x1D703}^{\prime }$ below so that all equations are written using the symbol  $\unicode[STIX]{x1D703}$ .

The meridional momentum equation is

(5.17) $$\begin{eqnarray}(hv)_{t}+(huv)_{x}+\left(hv^{2}+{\textstyle \frac{1}{2}}h^{2}(1+\unicode[STIX]{x1D703})\right)_{y}+{\textstyle \frac{1}{2}}yhu=-\unicode[STIX]{x1D6FE}hv,\end{eqnarray}$$

and the conservation form of the temperature equation is

(5.18) $$\begin{eqnarray}(h\unicode[STIX]{x1D703})_{t}+(hu\unicode[STIX]{x1D703})_{x}+(hv\unicode[STIX]{x1D703})_{y}=-\unicode[STIX]{x1D705}(h^{2}(1+\unicode[STIX]{x1D703})-h).\end{eqnarray}$$

The unit term from the decomposition $\unicode[STIX]{x1D6E9}=1+\unicode[STIX]{x1D703}$ disappears from the left-hand side of (5.18) because $h$ satisfies the mass conservation equation with no source terms. However, the combination $h^{2}(1+\unicode[STIX]{x1D703})$ appears in both the pressure gradient in (5.17) and the radiative relaxation term in (5.18). We may reduce this expression to a quadratic product by writing

(5.19) $$\begin{eqnarray}h^{2}\unicode[STIX]{x1D703}=h\unicode[STIX]{x1D703}+h^{\prime }\unicode[STIX]{x1D703}^{\prime }+O(a^{3}).\end{eqnarray}$$

The zonal mean of the temperature equation may thus be written as

(5.20) $$\begin{eqnarray}\langle \unicode[STIX]{x1D703}\rangle _{t}^{\ast }+\unicode[STIX]{x1D705}(\langle \unicode[STIX]{x1D703}\rangle ^{\ast }+\langle h\rangle -1)=-\langle v^{\prime }\unicode[STIX]{x1D703}^{\prime }\rangle _{y}-\unicode[STIX]{x1D705}(\langle h^{\prime }\unicode[STIX]{x1D703}^{\prime }\rangle +\langle h^{\prime 2}\rangle ).\end{eqnarray}$$

As before for the velocity components, we use $\langle \unicode[STIX]{x1D703}\rangle ^{\ast }$ instead of $\langle \unicode[STIX]{x1D703}\rangle$ to absorb a contribution from $\langle \unicode[STIX]{x1D703}^{\prime }h^{\prime }\rangle$ in the time derivative. As $\unicode[STIX]{x1D703}=O(a)$ under our decomposition, we may neglect factors of $\langle h\rangle =1+O(a^{2})$ that we could not neglect for $\langle \unicode[STIX]{x1D6E9}\rangle ^{\ast }$ . Similarly, the zonal mean of the meridional momentum equation is

(5.21) $$\begin{eqnarray}\langle v\rangle _{t}^{\ast }+{\textstyle \frac{1}{2}}y\langle u\rangle ^{\ast }+\unicode[STIX]{x1D6FE}\langle v\rangle ^{\ast }+\langle h\rangle _{y}+{\textstyle \frac{1}{2}}\langle \unicode[STIX]{x1D703}\rangle _{y}^{\ast }=-\langle v^{\prime }v^{\prime }\rangle _{y}-{\textstyle \frac{1}{2}}\langle h^{\prime 2}\rangle _{y}-{\textstyle \frac{1}{2}}\langle h^{\prime }\unicode[STIX]{x1D703}^{\prime }\rangle _{y},\end{eqnarray}$$

so we again have four equations for evolving $\langle h\rangle$ , $\langle u\rangle ^{\ast }$ , $\langle u\rangle ^{\ast }$ , $\langle \unicode[STIX]{x1D703}\rangle ^{\ast }$ with linear left-hand sides. The right-hand sides are again zonal means of quadratic products of the fluctuating quantities  $h^{\prime }$ , $u^{\prime }$ , $v^{\prime }$ , $\unicode[STIX]{x1D703}^{\prime }$ .

The meridional velocity perturbation for an equatorial Rossby wave is again $v^{\prime }=\unicode[STIX]{x1D6FC}\,\text{Re}\{\hat{v}(y)\exp (\text{i}(kx-\unicode[STIX]{x1D714}t))\}$ , with $\hat{v}(y)$ defined by (4.3) and (4.10). The corresponding perturbations to the thickness, temperature and zonal velocity take the same functional forms, with

(5.22) $$\begin{eqnarray}\left.\begin{array}{@{}c@{}}\displaystyle {\hat{h}}(y)=-\text{i}\left(\frac{\text{i}k\hat{u} +\text{d}\hat{v}/\text{d}y}{\unicode[STIX]{x1D714}}\right),\quad \hat{\unicode[STIX]{x1D6E9}}(y)=-\text{i}\left(\frac{\unicode[STIX]{x1D705}{\hat{h}}}{\unicode[STIX]{x1D714}+\text{i}\unicode[STIX]{x1D705}}\right),\\ \displaystyle \hat{u} (y)=\text{i}\left(\frac{k(1+\text{i}\unicode[STIX]{x1D705}/(2\unicode[STIX]{x1D714}))\,\text{d}\hat{v}/\text{d}y-{\textstyle \frac{1}{2}}y(\unicode[STIX]{x1D714}+\text{i}\unicode[STIX]{x1D705})\hat{v}}{k^{2}(1+\text{i}\unicode[STIX]{x1D705}/(2\unicode[STIX]{x1D714}))-(\unicode[STIX]{x1D714}+\text{i}\unicode[STIX]{x1D6FE})(\unicode[STIX]{x1D714}+\text{i}\unicode[STIX]{x1D705})}\right),\end{array}\right\}\end{eqnarray}$$

for $\unicode[STIX]{x1D714}\neq 0$ and $\unicode[STIX]{x1D714}\neq -\text{i}\unicode[STIX]{x1D705}$ . The vanishing denominator in the expression for $\hat{u}$ when $k^{2}=(\unicode[STIX]{x1D714}+\text{i}\unicode[STIX]{x1D6FE})(\unicode[STIX]{x1D714}+\text{i}\unicode[STIX]{x1D705})/(1+\text{i}\unicode[STIX]{x1D705}/(2\unicode[STIX]{x1D714}))$ again gives the dispersion relation for the equatorially trapped Kelvin waves. We choose the normalisation constant $\unicode[STIX]{x1D6FC}$ so that the disturbances $u^{\prime }$ , $v^{\prime }$ , $h^{\prime }$ and $\unicode[STIX]{x1D703}^{\prime }$ satisfy

(5.23) $$\begin{eqnarray}\frac{1}{2}\int \langle u^{\prime 2}\rangle +\langle v^{\prime 2}\rangle +\left\langle \left(h^{\prime }+\frac{1}{2}\unicode[STIX]{x1D703}^{\prime }\right)^{2}\right\rangle \,\text{d}y=a^{2}.\end{eqnarray}$$

The left-hand side is the dimensionless quadratic approximation for small-amplitude disturbances to the pseudoenergy

(5.24) $$\begin{eqnarray}\frac{1}{2}\int h(u^{2}+v^{2})+(h\sqrt{\unicode[STIX]{x1D6E9}}-h_{0}\sqrt{\unicode[STIX]{x1D6E9}_{0}})^{2}\,\text{d}x\,\text{d}y\end{eqnarray}$$

of finite-amplitude disturbances from a rest state of constant thickness $h_{0}$ and temperature $\unicode[STIX]{x1D6E9}_{0}$ in the thermal shallow water equations (Ripa Reference Ripa1995; Røed Reference Røed1997; Warneford & Dellar Reference Warneford and Dellar2013)

Figure 12.

The effective zonal force $-\langle u^{\prime }v^{\prime }\rangle _{y}$ on the right-hand side of (5.15) for $k=1$ , $n=1$ , and $a=1$ for scaling. Plot (a) shows dissipation solely by radiative relaxation ( $\unicode[STIX]{x1D705}=0.002,\unicode[STIX]{x1D6FE}=0$ ) and (b) shows dissipation solely by friction ( $\unicode[STIX]{x1D705}=0,\unicode[STIX]{x1D6FE}=0.001$ ). The zero line is shown dotted. The ratio of $\unicode[STIX]{x1D705}$ to $\unicode[STIX]{x1D6FE}$ reflects the symmetry of the dispersion relation in § 4.2.

Figure 13.

The effective zonal force $-\langle u^{\prime }v^{\prime }\rangle _{y}$ on the right-hand side of (5.15) for $k=1$ , $n=2$ , and $a=1$ for scaling. Plot (a) shows dissipation solely by radiative relaxation ( $\unicode[STIX]{x1D705}=0.002,\unicode[STIX]{x1D6FE}=0$ ) and (b) shows dissipation solely by friction ( $\unicode[STIX]{x1D705}=0,\unicode[STIX]{x1D6FE}=0.001$ ). The zero line is shown dotted. The ratio of $\unicode[STIX]{x1D705}$ to $\unicode[STIX]{x1D6FE}$ reflects the symmetry of the dispersion relation in § 4.2.

Figure 14.

The effective meridional forcing terms $-\langle v^{\prime }v^{\prime }\rangle _{y}$ , $-\langle h^{\prime 2}\rangle _{y}/2$ , $-\langle h^{\prime }\unicode[STIX]{x1D703}^{\prime }\rangle _{y}/2$ on the right-hand side of (5.21), and their sum, for dissipation solely by radiative relaxation ( $\unicode[STIX]{x1D705}=0.002,\unicode[STIX]{x1D6FE}=0$ ). All terms are calculated for $k=1$ , $a=1$ for scaling, and either (a)  $n=1$ or (b)  $n=2$ .

Figure 15.

The effective thermal terms $-\langle v^{\prime }\unicode[STIX]{x1D703}^{\prime }\rangle _{y}$ , $-\unicode[STIX]{x1D705}\langle h^{\prime }\unicode[STIX]{x1D703}^{\prime }\rangle$ , $-\unicode[STIX]{x1D705}\langle h^{\prime 2}\rangle$ on the right-hand side of (5.20), and their sum, for dissipation solely by radiative relaxation ( $\unicode[STIX]{x1D705}=0.002,\unicode[STIX]{x1D6FE}=0$ ). All terms are calculated for $k=1$ , $a=1$ for scaling, and either (a)  $n=1$ or (b)  $n=2$ .

Figures 12 and 13 show the Reynolds stress convergences on the right-hand side of (5.15). These all closely resemble those for the previous shallow water equations with radiative relaxation of the thickness. Thus the mean flow at the equator ( $y=0$ ) is almost identical to that for the previous model, provided $\unicode[STIX]{x1D705}$ is rescaled by a factor of 2. A perturbative analysis similar to that in § 4.1 shows that the Reynolds stress $\langle u^{\prime }v^{\prime }\rangle$ for this model is proportional to $\unicode[STIX]{x1D705}/2-\unicode[STIX]{x1D6FE}$ for small $\unicode[STIX]{x1D705}$ and  $\unicode[STIX]{x1D6FE}$ .

Figure 14 shows the forcing terms in the right-hand side of the mean meridional momentum equation (5.21). The dominant terms, $-\langle v^{\prime }v^{\prime }\rangle _{y}$ and $-\langle h^{\prime 2}\rangle _{y}/2$ are very similar to those in figure 11 for the shallow water model with relaxation of the thickness. The thermal term $-\langle h^{\prime }\unicode[STIX]{x1D703}^{\prime }\rangle _{y}/2$ is much smaller. Finally, figure 15 shows the forcing terms in the right-hand side of the mean temperature equation (5.20). These all vanish for dissipation solely by friction ( $\unicode[STIX]{x1D705}=0$ ), since the temperature perturbation $\hat{\unicode[STIX]{x1D703}}$ given by (5.22) then vanishes, so they are only shown for $\unicode[STIX]{x1D705}>0$ and $\unicode[STIX]{x1D6FE}=0$ . As for the meridional velocity, the contribution from $\langle h^{\prime }\unicode[STIX]{x1D703}^{\prime }\rangle$ is much smaller than the other contributions.