2.1.1 Annulus configuration

Figure 1.

Schematic of the differentially heated rotating annulus experiment. Temperatures $T_{a}$ and $T_{b}$ are prescribed at the inner ( $r=a$ ) and outer ( $r=b$ ) cylinder walls. $d$ represents the fluid height and $\unicode[STIX]{x1D6FA}$ the angular velocity of the system. A cylindrical coordinate system consisting of the azimuthal angle $\unicode[STIX]{x1D717}$ , the radial distance from the axis of rotation $r$ and the vertical distance from the bottom $z$ is defined. The dotted rectangular boxes indicate the regular cylindrical finite-volume grid. Courtesy of Borchert et al. (Reference Borchert, Achatz and Fruman2014).

A schematic of the differentially heated rotating annulus is shown in figure 1, where a cylindrical coordinate system consisting of the azimuthal angle $\unicode[STIX]{x1D717}$ , the radial distance from the axis of rotation $r$ and the vertical distance from the bottom $z$ is introduced. The experiment consists of two vertical coaxial cylinders. The inner cylinder at radius $r=a$ is cooled ( $T=T_{a}$ ) and the outer cylinder at radius $r=b$ is heated ( $T=T_{b}$ ). The tank is filled with water up to a height $d$ . Furthermore, the whole apparatus rotates at a specific angular velocity $\unicode[STIX]{x1D6FA}$ , forcing the fluid particles to experience Coriolis and centrifugal accelerations.

Table 1.

Physical parameters and derived dimensionless quantities for the atmosphere-like configuration developed in Borchert et al. (Reference Borchert, Achatz and Fruman2014).

The physical parameters used in this study are listed in table 1. Since deviations of the density $\unicode[STIX]{x0394}\hat{\unicode[STIX]{x1D70C}}$ from the constant background density $\hat{\unicode[STIX]{x1D70C}}_{0}$ at the reference temperature $T_{0}=(T_{a}+T_{b})/2$ are relatively small ( $|\unicode[STIX]{x0394}\hat{\unicode[STIX]{x1D70C}}|<0.01\hat{\unicode[STIX]{x1D70C}}_{0}$ ) in the considered temperature range, the dynamics of the system is described by the Boussinesq approximation. The pressure field is also divided up into a time-independent vertically varying background field $\hat{p}_{0}$ and a deviation $\unicode[STIX]{x0394}\hat{p}$ . The former is defined assuming hydrostatic equilibrium between the pressure gradient, gravity and the centrifugal acceleration, i.e.

where $\unicode[STIX]{x1D735}=\boldsymbol{e}_{\unicode[STIX]{x1D751}}(1/r)\unicode[STIX]{x2202}/\unicode[STIX]{x2202}\unicode[STIX]{x1D717}+\boldsymbol{e}_{\boldsymbol{r}}\unicode[STIX]{x2202}/\unicode[STIX]{x2202}r+\boldsymbol{e}_{\boldsymbol{z}}\unicode[STIX]{x2202}/\unicode[STIX]{x2202}z$ , $\boldsymbol{g}=-g\boldsymbol{e}_{\boldsymbol{z}}$ is the gravitational acceleration, $\unicode[STIX]{x1D734}=\unicode[STIX]{x1D6FA}\boldsymbol{e}_{\boldsymbol{z}}$ is the angular-velocity vector and $-\unicode[STIX]{x1D734}\times (\unicode[STIX]{x1D734}\times \boldsymbol{r})=\unicode[STIX]{x1D6FA}^{2}r\boldsymbol{e}_{\boldsymbol{r}}$ is the centrifugal acceleration. Here $\boldsymbol{e}_{\unicode[STIX]{x1D751}}$ , $\boldsymbol{e}_{\boldsymbol{r}}$ and $\boldsymbol{e}_{\boldsymbol{z}}$ are the azimuthal, radial and vertical unit vectors, forming a left-handed coordinate system in this order.

The momentum equation under the Boussinesq approximation can then be written as

where $p\equiv \unicode[STIX]{x0394}\hat{p}/\hat{\unicode[STIX]{x1D70C}}_{0}$ , $\unicode[STIX]{x1D70C}\equiv \unicode[STIX]{x0394}\hat{\unicode[STIX]{x1D70C}}/\hat{\unicode[STIX]{x1D70C}}_{0}$ and $\boldsymbol{v}=u\boldsymbol{e}_{\unicode[STIX]{x1D751}}+v\boldsymbol{e}_{\boldsymbol{r}}+w\boldsymbol{e}_{\boldsymbol{z}}$ is the velocity vector, $\unicode[STIX]{x1D644}$ is the unit tensor and $\unicode[STIX]{x1D748}$ represents the viscous stress tensor,

with the kinematic viscosity $\unicode[STIX]{x1D708}$ and the superscript $\text{T}$ indicating the transpose. The continuity equation is given by

and the thermodynamic energy equation and the equation of state are

where $T$ is the temperature, $\unicode[STIX]{x1D705}$ is the thermal diffusivity and $\unicode[STIX]{x1D70C}_{1}$ and $\unicode[STIX]{x1D70C}_{2}$ are fluid-dependent coefficients. The thermal diffusivity $\unicode[STIX]{x1D705}$ and the kinematic viscosity $\unicode[STIX]{x1D708}$ also depend on temperature and are fitted by parabolas

Strictly speaking, we use an extension of the original Boussinesq approximation since temperature dependence is incorporated in different fluid properties in addition to the density (Hignett et al. Reference Hignett, White, Carter, Jackson and Small1985). The values of the coefficients $\unicode[STIX]{x1D70C}_{1,2}$ , $\unicode[STIX]{x1D708}_{0,1,2}$ and $\unicode[STIX]{x1D705}_{0,1,2}$ are listed in table 1. As illustrated in Borchert et al. (Reference Borchert, Achatz, Remmler, Hickel, Harlander, Vincze, Alexandrov, Rieper, Heppelmann and Dolaptchiev.2015), these fits capture the fluid properties very well in the considered temperature range. As also shown there, for a smaller temperature difference between the annulus walls than considered in our study, the model shows good agreement with corresponding laboratory studies. Experimental validation of the appropriateness of the Boussinesq approximation in settings as considered here, with a comparatively large temperature difference, hence still seems highly desirable. Nonetheless, we do not expect that this approximation significantly influences the dynamics of the system. By scale analysis Gray & Giorgini (Reference Gray and Giorgini1976) have investigated under which conditions the traditional Boussinesq approximation holds, without temperature dependence in the fluid properties. Scrutiny of that study indicates that under the conditions considered here all non-Boussinesq extensions are probably negligibly small.

The physical parameters used in this study (see table 1) are taken from Borchert et al. (Reference Borchert, Achatz and Fruman2014). The configuration developed therein provides atmosphere-like conditions, since, in particular, the spatially averaged Brunt–Väisälä frequency $N$ is larger than the inertial frequency $f=2\unicode[STIX]{x1D6FA}$ . As explained in detail in § 3.1.1, the stratification $N$ is caused and maintained by the overturning circulation arising from the temperature gradient between the two side walls. This is why the temperature difference between top and bottom $\unicode[STIX]{x0394}T_{vert}$ is approximately equal to the radial temperature difference $T_{b}-T_{a}$ . Hence, $N$ can be estimated by

leading to $N/f=O(10)$ , which is approximately one order less than in the upper troposphere of midlatitudes (Esler & Polvani Reference Esler and Polvani2004). The importance of $N$ and $f$ in the theoretical description of IGWs becomes particularly clear when considering the dispersion relation of IGWs under Boussinesq approximation (Fritts & Alexander Reference Fritts and Alexander2003),

where $\hat{\unicode[STIX]{x1D714}}$ is the intrinsic frequency, $k$ , $l$ , $m$ are the zonal (azimuthal in the annulus context), meridional (negative radial) and vertical wavenumbers and $\unicode[STIX]{x1D6FC}=\text{arctan}(m/\sqrt{k^{2}+l^{2}})$ is the angle between the phase propagation and the horizontal plane. Thus, for a given value of $\hat{\unicode[STIX]{x1D714}}$ , $\unicode[STIX]{x1D6FC}$ is determined by the ratio of $N$ to $f$ . Moreover, the specific choice of parameters determines if the flow state will become baroclinically unstable, leading to the formation of baroclinic waves. As discussed in detail in Borchert et al. (Reference Borchert, Achatz, Remmler, Hickel, Harlander, Vincze, Alexandrov, Rieper, Heppelmann and Dolaptchiev.2015), a guideline for understanding the instability mechanism is provided by the quasi-geostrophic model developed by Eady (Reference Eady1949). Introducing the Rossby deformation radius $L_{d}=Nd/f$ and the dimensionless Burger number $Bu=L_{d}^{2}/(b-a)^{2}$ , the model predicts the flow to become unstable if

with $(\unicode[STIX]{x1D707}_{c}/\unicode[STIX]{x03C0})^{2}=0.583$ (Hide & Mason Reference Hide and Mason1975). Further dimensionless numbers used to describe the flow properties are provided by the Rossby number $Ro$ and the Ekman number $Ek$ . With $U$ and $L$ being characteristic horizontal velocity and length scales of the large-scale flow, the Rossby number

describes the ratio of the inertial force to the Coriolis force of a rotating fluid. As explained in § 2.2.1, this value can be used for a scale separation between large-scale balanced motions and IGWs (Vanneste Reference Vanneste2013). A rough estimate of $Ro$ can be obtained by inserting the thermal wind into the general definition (2.12), then yielding the thermal Rossby number (Borchert et al. Reference Borchert, Achatz, Remmler, Hickel, Harlander, Vincze, Alexandrov, Rieper, Heppelmann and Dolaptchiev.2015)

The Brunt–Väisälä frequency $N^{2}$ can be reinserted in the Rossby number, due to the equality of vertical and radial temperature differences in our configuration. The Rossby number thus happens to be equal to the Burger number of the flow. The Ekman number

represents the ratio of viscous forces to Coriolis force. The value is larger than the typical value observed in the actual atmosphere, since we only consider a thin layer of water moving at a relatively low velocity. However, the process of interest (IGW emission) seems to be largely unaffected by this, even though it impacts the subsequent dissipation.

2.1.2 Cartesian, doubly periodic configuration

Previous studies indicate that IGWs in the differentially heated rotating annulus originate both from the jet–front system and from the boundary layer located at the inner side wall of the annulus (Jacoby et al. Reference Jacoby, Read, Williams and Young2011; Randriamampianina Reference Randriamampianina2013; Borchert et al. Reference Borchert, Achatz and Fruman2014; Randriamampianina & Crespo del Arco Reference Randriamampianina and Crespo del Arco2015). We wish to contribute to the improvement of the physical understanding of the source processes, so we focus on IGWs originating from the jet–front system. Two supplementary approaches are therefore pursued here to quantify the role of this process, as compared to boundary-layer instabilities. In both approaches first the forcing of IGWs by the balanced part of the flow is identified, and then its effect is quantified using a tangent-linear model. In one set of investigations we stick to the annulus geometry where radial-boundary-layer processes always seem to matter. We also consider an idealised fluid-flow set-up without rigid side walls. This is achieved by adapting our model configuration to a Cartesian geometry with periodic boundary conditions in both horizontal directions. On the one hand we can demonstrate by this approach that an emission of IGWs by the jet–front system takes place, since side wall effects are excluded by construction. On the other hand we can quantify the contribution of both processes to the IGW field. For the doubly periodic configuration we introduce a left-handed Cartesian coordinate system represented by the unit vectors $\boldsymbol{e}_{\boldsymbol{x}}$ , $\boldsymbol{e}_{\boldsymbol{y}}$ and $\boldsymbol{e}_{\boldsymbol{z}}$ (with correspondences $\boldsymbol{e}_{\unicode[STIX]{x1D751}}\rightarrow \boldsymbol{e}_{\boldsymbol{x}}$ , $\boldsymbol{e}_{\boldsymbol{r}}\rightarrow \boldsymbol{e}_{\boldsymbol{y}}$ and $\boldsymbol{e}_{\boldsymbol{z}}=\boldsymbol{e}_{\boldsymbol{z}}$ ). The equation system to be solved is still given by (2.2)–(2.6), but the energy conservation equation (2.5) is modified to include a forcing term $F$ causing a baroclinically unstable hyperbolic tangent temperature profile

with

and

where $T_{a}$ and $T_{b}$ are the minimum and maximum forced temperatures ( $T_{b}>T_{a}$ ) and $\unicode[STIX]{x1D70E}_{y}$ is the relative thickness of the temperature jump in the forcing. $\unicode[STIX]{x1D70F}$ is a relaxation time which modulates the amplitude of the temperature forcing. Choosing $1/\unicode[STIX]{x1D70F}=1/1000~\text{s}$ turns out to affect the temperature profile appropriately. Moreover, this choice ensures that the relaxation acts on time scales well above the dynamical time scales of interest in the spontaneous emission process. Figure 2 displays the hyperbolic temperature profile $T_{r}$ and the values of its parameters and the corresponding dimensionless quantities can be found in table 2. The domain size is chosen such that we expect it to accommodate one wavelength of the baroclinic wave. Beyond the Cartesian geometry without solid horizontal walls, however, the set-up is still very close to the annulus set-up: water is used as working fluid with the same viscous and diffusive properties as described in § 2.1.1.

Figure 2.

Tangent hyperbolic temperature profile $T_{r}(y)$ as implemented in the Cartesian, doubly periodic model configuration. See expression (2.17) for the functional dependence.

Table 2.

Physical parameters and derived dimensionless quantities for the doubly periodic Cartesian configuration.

2.1.3 Numerical methods

The numerical model used for the simulations of the differentially heated rotating annulus is the same as used by Borchert et al. (Reference Borchert, Achatz and Fruman2014, Reference Borchert, Achatz, Remmler, Hickel, Harlander, Vincze, Alexandrov, Rieper, Heppelmann and Dolaptchiev.2015). It makes use of a finite-volume algorithm to solve the equation system (2.2)–(2.6) numerically. Therefore, the annulus volume is subdivided into volume cells of azimuthal width $\unicode[STIX]{x0394}\unicode[STIX]{x1D717}$ , radial width $\unicode[STIX]{x0394}r$ and vertical extension $\unicode[STIX]{x0394}z$ and the variables are discretised on a staggered C-grid (Arakawa & Lamb Reference Arakawa and Lamb1977). In contrast to the studies of Borchert et al. (Reference Borchert, Achatz and Fruman2014, Reference Borchert, Achatz, Remmler, Hickel, Harlander, Vincze, Alexandrov, Rieper, Heppelmann and Dolaptchiev.2015) in which the implicit subgrid-scale parametrisation ALDM (adaptive local deconvolution method, Hickel, Adams & Domaradzki Reference Hickel, Adams and Domaradzki2006) is implemented through a special handling of the advective terms, we determine the fluxes from the surrounding volume-averaged velocities using the second-order centred-difference option of the model. This is justified since our numerical simulations are carried out with a relatively high spatial resolution allowing explicit resolution of most of the small-scale processes. The time integration of the four prognostic equations for the velocity fields $\boldsymbol{v}=(u,v,w)$ and the temperature $T$ is done using a low-storage third-order Runge–Kutta method (Williamson Reference Williamson1980) with an adaptive time step determined by the instantaneous velocity field. The pressure field $p$ is diagnosed from the three velocity components by solving a Poisson equation which ensures that the continuity equation (2.4) is satisfied.

2.2.1 Balanced and unbalanced flow parts and their interaction

In order to examine the characteristics and sources of IGWs as precisely as possible, the extraction of the IGW signal from the remaining flow is essential. One possible way to do that is presented in Borchert et al. (Reference Borchert, Achatz and Fruman2014) who determine the large-scale part of the flow by a moving average. The difference between the full and the averaged field is then associated with the small-scale part including the IGW signal. Furthermore, a linear modal decomposition is applied to estimate the contribution of the IGWs to the small-scale structures of the flow. Here, we choose another definition for the separation of the flow into a balanced (subscript $b$ ) and unbalanced (subscript $u$ ) part

The first term on the right-hand side satisfies appropriate balance relations (e.g. the geostrophic and the hydrostatic balance) and can be determined from the left-hand side state by potential vorticity inversion (e.g. Davis & Emanuel Reference Davis and Emanuel1991). The unbalanced flow is given by the difference between the full and the balance fields and contains the IGWs. Note that we introduced the buoyancy $B=-\unicode[STIX]{x1D70C}g$ , and that the pressure $p$ and the buoyancy $B$ represent deviations from the mean thermal stratification $\hat{p}_{0}=\hat{p}_{0}(z)$ and $\hat{B}_{0}=\text{d}\hat{p}_{0}/\text{d}z$ . The introduction of this separation also enables investigation of the coupling between the large-scale balanced part and the IGWs. Many different definitions can be found in the literature to determine the balanced part of the flow ranging from rather simple balance assumptions to more complex higher-order balance approaches (e.g. Warn et al. Reference Warn, Bokhove, Shepherd and Vallis1995; Zhang et al. Reference Zhang, Koch, Davis and Kaplan2000; Viúdez & Dritschel Reference Viúdez and Dritschel2006; Snyder et al. Reference Snyder, Plougonven and Muraki2009; Wang & Zhang Reference Wang and Zhang2010). In general, the balanced part of the flow should capture the large-scale motion of the stratified fluid (i.e. the baroclinic waves) as precisely as possible.

In most models potential vorticity is assumed to be a key variable of the large-scale horizontal flow to which the balanced part of the flow shall be the only contribution by definition. After computing the PV from the full flow state, the balanced part of the flow is computed diagnostically from the PV using balance relations (PV inversion, e.g. Hoskins et al. Reference Hoskins, McIntyre and Robertson1985; McIntyre & Norton Reference McIntyre and Norton2000; Vanneste Reference Vanneste2013). Based on theoretical considerations, but also for practical purposes, we take a slightly different route here. In our configuration of the differentially heated rotating annulus (see § 2.1) the Rossby number is small ( $Ro<1$ ) in most locations. As shown by Bühler & McIntyre (Reference Bühler and McIntyre2005) in the Lagrangian mean and by Achatz et al. (Reference Achatz, Ribstein, Senf and Klein2017) in the Eulerian perspective, in that limit IGWs contribute to the nonlinear part of PV, while the linear part is determined exclusively by a geostrophically and hydrostatically balanced component, as also in quasi-geostrophic theory (Charney Reference Charney1948; Pedlosky Reference Pedlosky1987; Vallis Reference Vallis2006). Moreover, as can be verified from their polarisation relations, linear IGWs have no linear PV (Phillips Reference Phillips1963; Mohebalhojeh & Dritschel Reference Mohebalhojeh and Dritschel2001; Smith & Waleffe Reference Smith and Waleffe2002). Hence we define a so-called balanced flow so that it satisfies the geostrophic and hydrostatic balance relations

with the horizontal velocity $\boldsymbol{u}_{\boldsymbol{b}}$ , and so that it yields the total linear PV. Consequently, we have

Herein, $\unicode[STIX]{x1D701}=\boldsymbol{e}_{\boldsymbol{z}}\boldsymbol{\cdot }(\unicode[STIX]{x1D735}_{\boldsymbol{h}}\times \boldsymbol{u})$ represents the vertical component of the vorticity, $\unicode[STIX]{x1D735}_{\boldsymbol{h}}$ is the horizontal part of the nabla operator, $\unicode[STIX]{x1D701}_{b}=\boldsymbol{e}_{\boldsymbol{z}}\boldsymbol{\cdot }(\unicode[STIX]{x1D735}_{\boldsymbol{h}}\times \boldsymbol{u}_{\boldsymbol{b}})$ is the balanced vertical vorticity and $\unicode[STIX]{x1D6FB}_{qg}^{2}\equiv (\unicode[STIX]{x1D6FB}_{h}^{2}+f^{2}/N^{2}\unicode[STIX]{x2202}^{2}/\unicode[STIX]{x2202}z^{2})$ is the quasi-geostrophic Laplacian. Naturally this also implies that the remainder of the flow (henceforth called the unbalanced part, as it contains the IGW part and all of the imbalance, in addition to a usually weaker balanced contribution), $\boldsymbol{u}_{\boldsymbol{u}}=\boldsymbol{u}-\boldsymbol{u}_{\boldsymbol{b}}$ and $B_{u}=B-B_{b}$ , does not contribute to linear PV

where $\unicode[STIX]{x1D701}_{u}=\boldsymbol{e}_{\boldsymbol{z}}\boldsymbol{\cdot }(\unicode[STIX]{x1D735}_{\boldsymbol{h}}\times \boldsymbol{u}_{\boldsymbol{u}})$ is the unbalanced vertical vorticity. To avoid misunderstandings, we also note that the balanced pressure $p_{b}$ that we obtain from inverting linear PV in (2.21) is only approximately identical to the geostrophic pressure, i.e. the leading-order part in a Rossby-number expansion of the total pressure fluctuations. As shown by Muraki, Snyder & Rotunno (Reference Muraki, Snyder and Rotunno1999) for the hydrostatic case, e.g. the next-order corrections to the balanced flow, that one might term the ageostrophic flow, do enter linear PV, so that ageostrophic linear PV does not vanish. However, these next-order corrections are smaller by $O(Ro)$ , so that the (non-vanishing) deviations between balanced flow, as defined here, and the geostrophic flow are small in the limit of small $Ro$ . This then does not outweigh the practical advantage we have from the linear flow decomposition just described, as the latter allows a straightforward reformulation of the dynamics in terms of an interaction between balanced flow and unbalanced (IGW carrying) flow, as described below.

Nonetheless, as what we call unbalanced flow does contain some balanced contributions if the Rossby number is sufficiently large (e.g. McWilliams Reference McWilliams1985; Muraki et al. Reference Muraki, Snyder and Rotunno1999), our diagnostics below will go a step further by extracting the contributions from unbalanced flow balanced to next order in the Rossby number, using the quasi-geostrophic omega equation. Moreover, in order to validate the assumption that the unbalanced part of the flow mainly consists of IGWs, we have also applied a linear modal decomposition of the balanced and unbalanced flow, as described in Borchert et al. (Reference Borchert, Achatz and Fruman2014). The amounts of energy contained in the geostrophic and in the IGW modes (not shown) do indeed confirm that the unbalanced part of the flow is dominated by IGWs since the geostrophic energy in the unbalanced flow is approximately one order of magnitude less in amplitude than the IGW energy. By the same means we have also convinced ourselves that the balanced flow does not contain any IGW energy at all.

For an investigation of the interaction between balanced and unbalanced flow, we follow Borchert et al. (Reference Borchert, Achatz and Fruman2014) and simplify the dynamical equations by neglecting friction, heat conduction and centrifugal acceleration. Furthermore, we replace the thermodynamic energy equation by an equivalent equation for the buoyancy. The resulting Boussinesq system is (Vallis Reference Vallis2006)

where $\text{D}/\text{D}t=\unicode[STIX]{x2202}/\unicode[STIX]{x2202}t+\boldsymbol{v}\boldsymbol{\cdot }\unicode[STIX]{x1D735}$ is the material derivative. As shown in appendix A, one can derive from these equations a prognostic equation for the (balanced) linear PV, and hence for the balanced flow,

Note that $\text{D}\unicode[STIX]{x1D6F1}/\text{D}t$ has no purely balanced contribution since $\unicode[STIX]{x1D6FF}=\unicode[STIX]{x1D6FF}_{u}$ (see (2.60)), $w=w_{u}$ and $(\unicode[STIX]{x2202}\boldsymbol{u}_{\boldsymbol{b}}/\unicode[STIX]{x2202}z)\boldsymbol{\cdot }\unicode[STIX]{x1D735}_{\boldsymbol{h}}B_{b}=0$ . Therefore, $\unicode[STIX]{x1D6F1}$ is conserved in the absence of an unbalanced flow component (i.e. when $\boldsymbol{u}=\boldsymbol{u}_{\boldsymbol{b}}$ and $B=B_{b}$ ).

To obtain a set of prognostic equations for the unbalanced flow, we now decompose the variables into a geostrophically and hydrostatically balanced and an unbalanced part as described in § 2.2.1. Additionally, we make use of the geostrophic and hydrostatic balance relations (2.19), (2.20) and the fact that the unbalanced flow, as defined here, has zero linear PV (2.22). This leads to the prognostic system

Here, the material derivatives of the balanced fields on the right-hand side, both actually functions of the balanced and unbalanced flow components, have been further subdivided into a purely balanced and an unbalanced part. As shown in appendix B, these are

with

and

with

Herein, $\unicode[STIX]{x1D6FB}_{qg}^{-2}$ denotes the inversion of the quasi-geostrophic Laplacian defined in (2.21) and $\unicode[STIX]{x1D735}_{\boldsymbol{q}\boldsymbol{g}}\equiv \unicode[STIX]{x1D735}_{\boldsymbol{h}}+\boldsymbol{e}_{\boldsymbol{z}}(f/N)\unicode[STIX]{x2202}/\unicode[STIX]{x2202}z$ (provided that $N^{2}>0$ ) such that $\unicode[STIX]{x1D6FB}_{qg}^{2}=\unicode[STIX]{x1D735}_{\boldsymbol{q}\boldsymbol{g}}\boldsymbol{\cdot }\unicode[STIX]{x1D735}_{\boldsymbol{q}\boldsymbol{g}}$ . In addition, $\boldsymbol{\cdot }\boldsymbol{\cdot }$ denotes the double scalar product of two tensors (definition in appendix B). We have thus obtained a rigorous reformulation of the Boussinesq system (2.23)–(2.26) in terms of an interaction between balanced (geostrophic and hydrostatic) and unbalanced flow parts. Of central importance for the further investigations is the observation that even in the initial absence of any unbalanced flow part, and hence also IGWs, the balanced part of the material derivative of the balanced pressure in (2.35) can be non-zero, as well as the balanced material derivatives of balanced wind (2.33) and balanced buoyancy (2.34). This is a direct forcing of the unbalanced flow in (2.28) and (2.30) by the balanced (geostrophic and hydrostatic) flow!

2.2.2 Boundary conditions for the inversion problems

The dynamical decomposition described above entails various inversions that are only well defined with the corresponding boundary conditions, and also turn out to be very sensitive to them. These are non-trivial in the non-periodic directions. To begin with, the linear PV (2.21) is to be inverted to obtain the balanced pressure $p_{b}$ , which is then used to obtain the balanced horizontal velocity field $\boldsymbol{u}_{\boldsymbol{b}}$ and the buoyancy distribution $B_{b}$ using (2.19) and (2.20). The corresponding radial boundary condition in the annulus set-up is obtained from assuming zero azimuthal balanced flow at the side walls. Due to the geostrophic balance (2.19) the radial balanced pressure gradient must vanish,

The vertical boundary condition, both for the annulus and for the doubly periodic Cartesian set-up, is obtained from hydrostatic equilibrium (Zhang et al. Reference Zhang, Koch, Davis and Kaplan2000)

For the solution of (2.35) and (2.38) we obtain the vertical boundary condition

Here, we make use of the hydrostatic equilibrium (2.20) and the buoyancy equation (2.25), together with $w(z=0)=w(z=d)=0$ . Next we separate the terms into a balanced and an unbalanced part,

Analogously, the radial boundary conditions, only for the annulus set-up, can be derived as

where again the azimuthal balanced velocity $u_{b}=1/f(\unicode[STIX]{x2202}p_{b}/\unicode[STIX]{x2202}r)$ vanishes at the radial boundaries. Finally we obtain

All operator inversions are done using a preconditioned biconjugate gradient stabilised (BiCGSTAB) method (Van der Vorst Reference Van der Vorst1992).

2.2.3 A tangent-linear model to describe the unbalanced flow

For a systematic investigation of the balanced forcing of the unbalanced flow, and how much it contributes to the IGW emission in the differentially heated rotating annulus, we make use of a tangent-linear model to simulate the dynamics of the unbalanced part of the flow on the geostrophically and hydrostatically balanced background. Such an approach seems reasonable when the amplitudes of the unbalanced flow are sufficiently small and nonlinear self-interactions can be neglected over a certain integration period. Snyder et al. (Reference Snyder, Plougonven and Muraki2009) and Wang & Zhang (Reference Wang and Zhang2010) have already used tangent-linear models to study the spontaneous IGW emission in a vortex dipole. In their studies the forced linear model simulations compare well with the IGWs signal seen in the fully nonlinear model. In this section we present our tangent-linear approach, compare it to those used by others and finally summarise some technical details. To clarify our concept, we rewrite the annulus equations (2.27)–(2.32) as follows:

where $\boldsymbol{s}_{\boldsymbol{b}}(\boldsymbol{x},t)$ and $\boldsymbol{s}_{\boldsymbol{u}}(\boldsymbol{x},t)$ are the balanced and unbalanced prognostic variables, and $\boldsymbol{G}_{\boldsymbol{b}}(\boldsymbol{s}_{\boldsymbol{b}},\boldsymbol{s}_{\boldsymbol{u}})$ is the nonlinear tendency of the balanced flow. The nonlinear tendency $\boldsymbol{G}_{\boldsymbol{u}}(\boldsymbol{s}_{\boldsymbol{b}},\boldsymbol{s}_{\boldsymbol{u}})$ of the unbalanced flow is decomposed into a forcing $\boldsymbol{F}_{\boldsymbol{u}}(\boldsymbol{s}_{\boldsymbol{b}})$ depending only on the balanced flow, a linear part with an operator $\boldsymbol{L}_{\boldsymbol{u}}(\boldsymbol{s}_{\boldsymbol{b}})$ and the nonlinear unbalanced self-interaction terms $\boldsymbol{N}_{\boldsymbol{u}}(\boldsymbol{s}_{\boldsymbol{u}})$ that are all quadratic in the unbalanced variables. In our tangent-linear model we prescribe the balanced flow from an exact solution of the nonlinear annulus equations, and assume sufficiently weak unbalanced amplitudes in $\boldsymbol{s}_{\boldsymbol{u}}$ so that $\boldsymbol{N}_{\boldsymbol{u}}$ can be neglected. This leads to the linear prognostic system

Since each term on the right-hand side contributes additively to the temporal evolution of the unbalanced part $\boldsymbol{s}_{\boldsymbol{u}}$ , causes and effects can be assigned more easily to each other than would be possible in the fully nonlinear system. This enables us to explicitly quantify how much the balanced forcing contributes to unbalanced tendencies. Both terms on the right-hand side in (2.47) are important influencing factors for the evolution of a wave packet. Their different roles can be understood by considering the linear system as a simple forced oscillator (Plougonven & Zhang Reference Plougonven and Zhang2014). $\boldsymbol{F}_{\boldsymbol{u}}$ forces a range of frequencies/wavelengths and controls the amplitude of $\boldsymbol{s}_{\boldsymbol{u}}$ . In contrast, the linear operator $\boldsymbol{L}_{\boldsymbol{u}}$ mainly influences the structure of the wave packet by affecting the location, orientation and frequency of the wave. This becomes particularly clear when the linear model is initialised with zero unbalanced part and only the (generally non-zero) term $\boldsymbol{F}_{\boldsymbol{u}}(\boldsymbol{s}_{\boldsymbol{b}})$ leads to a forcing of the unbalanced flow and thus may induce IGWs.

The tangent-linear model is obtained by linearising the equation system (2.28)–(2.32), with prescribed time-dependent balanced flow as observed in the nonlinear model integrations. Most of the terms already show a tangent-linear structure. Only the material derivatives on the left-hand side and the material derivative of the PV (2.27), used in (2.38), are affected by the linearisation leading to

where

with

and

These equations form a closed forced system for the linear unbalanced flow components. Note that in the limit of small Rossby number, small IGW scales and small aspect ratio, the wave equation of Plougonven & Zhang (Reference Plougonven and Zhang2007) could be derived from this system. However, such a step is not necessary for the numerical integration of our model, and we instead keep it in the present more general form.

It seems worthwhile comparing our approach to related studies by others. Plougonven & Zhang (Reference Plougonven and Zhang2007) suggested a forcing of the unbalanced flow that corresponds to the total material derivatives of the balanced flow on the right-hand sides of (2.28) and (2.30). This approach was also used by Wang & Zhang (Reference Wang and Zhang2010). However, these material derivatives have an unbalanced part that is due to the unbalanced flow and belongs to the linear operator. This problem was avoided by Snyder et al. (Reference Snyder, Plougonven and Muraki2009) who obtained a linear model by prescribing a time-dependent balanced flow from the integration of a quasi-geostrophic model. This would correspond to integrating (2.27) with zero right-hand side. The linear equations are then forced by the residual tendency difference between the time tendencies of the quasi-geostrophic solution and the full primitive equations. In contrast to this approach, we use the real time-dependent balanced flow as results from the full dynamics. To extract the part of the nonlinear forcing (subscript $b$ in the material derivatives on the right-hand side in (2.48)–(2.52)) which is exclusively due to the balanced flow, we have to take the steps described above. This enables more direct exploration of the role of the balanced flow in the generation and propagation of the IGWs which are included in the unbalanced flow.

For consistency, the viscous friction and the heat conduction appearing in the full system (2.2)–(2.6) are also implemented in the tangent-linear equations. We have also performed sensitivity tests changing the values of $\unicode[STIX]{x1D708}$ and $\unicode[STIX]{x1D705}$ in order to investigate their influence on the linear dynamics and to test their ability as simplified parametrisation of the neglected nonlinearities. However, it turns out that no modifications from the original values (see (2.7) and (2.8)) are required for the integration time periods we are interested in. Numerically, the quasi-geostrophically and hydrostatically balanced part is calculated each time step from the full fields (see § 2.2.1). Therefore, and for all following considerations, $N^{2}$ is assumed to be constant throughout the domain allowing an efficient model performance. This is justifiable, since the actual vertical profile of $N^{2}$ does not depart strongly from a uniform value in the region of interest ( $0.5<z<3.5~\text{cm}$ ). The balanced fields serve as background for the tangent-linear model which is integrated parallel to the full model. However, some tests show that, independently of the initial condition, the linear unbalanced flow components diverge after a few seconds of integration time. Further investigations point out very fast growth rates at the inner and outer cylinders of the annulus, most likely caused by boundary-layer instabilities. In order to suppress these growth rates at the side walls, the linear fields are multiplied by a window function

at each time step. Therein, $L_{s}=\unicode[STIX]{x1D6FE}(1-\unicode[STIX]{x1D6FD})L_{y}$ and $\unicode[STIX]{x1D6FD}$ , $\unicode[STIX]{x1D6FE}$ are freely selectable tuning parameters. Varying these values does not influence the occurrence of the IGWs. Therefore, we decided to use $\unicode[STIX]{x1D6FD}=0.95$ and $\unicode[STIX]{x1D6FE}=1.0$ , whereby only grid points located near the side walls are affected. Note that for simplicity reasons we shift the origin of the radial coordinate to the radius $r^{\star }=r-(a+b)/2$ before applying (2.57) on the fields. In general, this modification has two major impacts. On the one hand the wave emission in the internal region and, in particular, in the vicinity of the jet exit region might be influenced causing a reduced correlation between the full and the linear model fields. However, on the other hand the application of the window function also inhibits the generation of IGWs at the boundaries as seen in Jacoby et al. (Reference Jacoby, Read, Williams and Young2011) and Randriamampianina & Crespo del Arco (Reference Randriamampianina and Crespo del Arco2015). As a consequence, we can further concentrate on the part of the IGW signal resulting from spontaneous imbalances rather than from boundary-layer instabilities. Obviously, such a window function is not to be used in the doubly periodic Cartesian set-up.

2.2.4 Balanced vertical velocity and horizontal divergence from the quasi-geostrophic omega equation

The separation into geostrophic/hydrostatic and unbalanced flow entails that horizontal velocity divergence and vertical velocity are, to leading order in $Ro$ , exclusively due to the unbalanced flow. However, if we include terms of next order in $Ro$ into the definition of the balanced flow part, the horizontal divergence has a non-vanishing balanced part $\unicode[STIX]{x1D6FF}_{bal}$ (as is known from the quasi-geostrophic theory). This part can be diagnosed from the QG omega equation (Hoskins, Draghici & Davies Reference Hoskins, Draghici and Davies1978; Holton Reference Holton2004; Danioux et al. Reference Danioux, Vanneste, Klein and Sasaki.2012)

with the boundary conditions $w_{bal}(z=0)=w_{bal}(z=d)=0$ . $\boldsymbol{Q}=\unicode[STIX]{x1D735}_{\boldsymbol{h}}\boldsymbol{u}_{\boldsymbol{b}}\boldsymbol{\cdot }\unicode[STIX]{x1D735}_{\boldsymbol{h}}B_{b}$ depends only on the balanced fields. The result is then used to estimate the balanced part

of the total horizontal velocity divergence

As this next-order definition of the balanced flow part is widely used in the literature, we will use $\unicode[STIX]{x1D6FF}_{unbal}=\unicode[STIX]{x1D6FF}-\unicode[STIX]{x1D6FF}_{bal}$ as an indicator for IGWs in the following sections. Corresponding imbalance diagnosis is equivalent to the use of the nonlinear balance residual of the divergence equation, as suggested by Zhang (Reference Zhang2004). To see this one can refer to the expansions given by McWilliams (Reference McWilliams1985). There it is shown that the nonlinear balance residual in the divergence equation vanishes to $O(Ro^{2})$ if Rossby and Froude number are of the same order, as is the case in quasi-geostrophic theory. The balanced divergence from the quasi-geostrophic omega equation is exact to $O(Ro)$ so that the corresponding divergence residual also indicates imbalance to $O(Ro^{2})$ . Finally we also note that the error we introduce by using in $\boldsymbol{Q}$ the balanced flow, given by $\boldsymbol{u}_{\boldsymbol{b}}$ and $B_{b}$ , instead of the geostrophic flow, in the sense of the leading-order expansion of all variables in terms of the $Ro$ , is by $O(Ro)$ smaller than $\unicode[STIX]{x1D6FF}_{bal}$ .