2.1. Two-layer model equations

We study a two-layer QG model on a $\beta$ -plane, over a linear bottom slope, with linear bottom friction and forced externally at the surface (e.g. Pedlosky Reference Pedlosky1987; Leng & Bai Reference Leng and Bai2018). We index the upper layer with $i=1$ and the lower layer with $i=2$ . The two layers have densities $\rho _i$ and thicknesses $H_i$ ; the total depth is $H \equiv H_1 + H_2$ . The total flow field is the sum of a uniform and stationary background field and an eddy field that varies in space and time. We assume that the eddy field is of small amplitude compared to the background field. The background PV is denoted by $Q_{i}$ , and the background flow velocity vector by $\boldsymbol{U}_i$ . Finally, the eddy PV, eddy flow velocities and eddy streamfunction are denoted by $q_i$ , $\boldsymbol {u}_i$ and $\psi _i$ , respectively. The QG PV equations for the two-layer system are

(2.1) \begin{align} \frac {\partial q_i}{\partial t} + \left (\boldsymbol{U}_i + \boldsymbol{u}_i \right ) \boldsymbol{\cdot} \boldsymbol{\nabla} \left (Q_i+q_i\right ) = \delta _{i1}\mathcal{F} -\delta _{i2}\mu\, \nabla ^2 \psi _2, \end{align}

where $\delta _{ij}$ is the Kronecker delta function, $\mathcal{F}$ is the surface forcing, and $\mu$ is the inverse frictional time scale. The eddy PV and velocities are related to the eddy streamfunctions:

(2.2a) \begin{align} &q_i = \nabla ^2 \psi _i + (-1)^i F_i \left ( \psi _1 - \psi _2\right ), \end{align}

(2.2b) \begin{align} &\,\,\quad \boldsymbol{u}_i = \left (-\partial \psi _i/\partial y,\partial \psi _i/\partial x\right ). \end{align}

Here, $F_i$ is the square of the inverse deformation radius in layer $i$ , given by

(2.3) \begin{align} F_i = \frac {f_0^2}{g'H_i}, \quad g' = g \frac {\rho _2 - \rho _1}{\rho _2}, \end{align}

with $f_0$ the Coriolis parameter, and $g'$ the reduced gravity. We write the background velocity in the upper layer as $\boldsymbol{U}_1 = (U_1,V_1 )$ , and in the lower layer as $\boldsymbol{U}_2 = (U_2,V_2 )$ . The vertical shears of the zonal and meridional background flow are $\Delta U = U_1 - U_2$ and $\Delta V = V_1 - V_2$ , respectively. The (linear) bottom slope $\boldsymbol{\alpha } = (\alpha _x, \alpha _y )$ induces a topographic PV gradient $\boldsymbol{\nabla} B = ({f_0}/{H_2})\boldsymbol{\alpha }$ in the lower layer. The total background PV gradient is the sum of the stretching, planetary and topographic PV gradients:

(2.4a) \begin{align} &\qquad \boldsymbol{\nabla} Q_1 = \left (-F_1\, \Delta V, F_1\, \Delta U + \beta \right ), \end{align}

(2.4b) \begin{align} &\boldsymbol{\nabla} Q_2 = \left (F_2\, \Delta V + B_x, -F_2\, \Delta U + \beta + B_y \right ). \end{align}

Finally, the term $\boldsymbol{u}_i \boldsymbol{\cdot} \boldsymbol{\nabla} q_i$ in (2.1) represents nonlinear eddy–eddy interactions, which we will ignore in our linear analysis. Thus the linearised versions of (2.1) in terms of the eddy streamfunctions are

(2.5a) \begin{align} &\qquad \left (\frac {\partial }{\partial t} + U_1 \frac {\partial }{\partial x} + V_1 \frac {\partial }{\partial y} \right ) \bigl(\nabla ^2 \psi _1 + F_1 (\psi _2 - \psi _1 ) \bigr) + \beta V_1 + F_1 \left (U_1V_2 - U_2V_1\right ) \notag\\&\qquad\quad + \left (F_1\, \Delta U + \beta \right )\frac {\partial \psi _1}{\partial x} + F_1\, \Delta V\, \frac {\partial \psi _1}{\partial y} = \mathcal{F}, \end{align}

(2.5b) \begin{align} &\left (\frac {\partial }{\partial t} + U_2 \frac {\partial }{\partial x} + V_2 \frac {\partial }{\partial y} \right ) \bigl(\nabla ^2 \psi _2 + F_2 (\psi _1 - \psi _2 ) \bigr) + B_x U_2 + \left (\beta + B_y \right )V_2 \notag\\&\quad + F_2 \left (U_2V_1 - U_1V_2\right ) + \left (-F_2\, \Delta U + \beta + B_y \right )\frac {\partial \psi _2}{\partial x} - \left (F_2\, \Delta V + B_x \right )\frac {\partial \psi _2}{\partial y} = -\mu\, \nabla ^2\psi _2. \end{align}

The lowest-order balance in (2.5a ) is the generalised Sverdrup balance (Sverdrup Reference Sverdrup1947), by which surface forcing ( $\mathcal{F}$ ) permits the mean flow to cross the mean PV gradient. There is no forcing in (2.5b ), implying that the mean flow must be parallel to the PV contours. This requires that

(2.6) \begin{equation} \frac {U_2}{V_2} = \frac {F_2 U_1 - \beta - B_y}{F_2 V_1 + B_x} \,. \end{equation}

Thus $U_2$ and $V_2$ are not independent. We will retain them for now, for generality, but will focus subsequently on the case $U_2=V_2=0$ .

At next order, (2.5a ) and (2.5b ) become homogeneous equations for the eddy streamfunctions $\psi _i$ (Kamenkovich & Pedlosky Reference Kamenkovich and Pedlosky1996; Leng & Bai Reference Leng and Bai2018). These are the equations on which we will focus:

(2.7a) \begin{align} &\left ( \! \frac {\partial }{\partial t} + U_1 \frac {\partial }{\partial x} + V_1 \frac {\partial }{\partial y} \! \right ) \bigl(\nabla ^2 \psi _1 + F_1 (\psi _2 - \psi _1 )\bigr) + \left (F_1\, \Delta U + \beta \right )\! \frac {\partial \psi _1}{\partial x} + F_1\, \Delta V \frac {\partial \psi _1}{\partial y} = 0, \end{align}

(2.7b) \begin{align} &\quad\left (\frac {\partial }{\partial t} + U_2 \frac {\partial }{\partial x} + V_2 \frac {\partial }{\partial y} \right ) \bigl(\nabla ^2 \psi _2 + F_2 (\psi _1 - \psi _2 ) \bigr) + \left (-F_2\, \Delta U + \beta + B_y \right )\frac {\partial \psi _2}{\partial x} \notag\\&\qquad - \left (F_2\, \Delta V + B_x \right )\frac {\partial \psi _2}{\partial y} = -\mu\, \nabla ^2\psi _2. \end{align}

2.2. Dispersion relation for wave solutions

We look for plane wave solutions of the eddy streamfunctions in a doubly periodic domain:

(2.8) \begin{align} \psi _{1,2} = \hat {\psi }_{1,2} e^{{i}({kx+ly-\sigma t})}, \end{align}

where $\hat {\psi }_{1,2}$ denotes the wave amplitude, $\mathbf{K} =(k,l) = \kappa (\cos \theta , \sin \theta )$ is the wavenumber vector, and $\sigma$ is the angular frequency. The wave phase speed magnitude $c$ is given by $c=\sigma /\kappa$ . Substituting (2.8) into (2.7) yields a matrix equation for the wave amplitudes:

(2.9) \begin{align} \begin{pmatrix} M_{11} & M_{12} \\ M_{21} & M_{22} \end{pmatrix} \begin{pmatrix} \hat {\psi }_1 \\ \hat {\psi }_2 \end{pmatrix} = 0, \end{align}

with

(2.10a) \begin{align} &\qquad\qquad M_{11} = c\kappa \bigl(\kappa ^2 + F_1 \bigr) -\kappa ^2 \left (kU_1+lV_1\right ) - F_1 \left (kU_2+lV_2\right ) + k\beta , \end{align}

(2.10b) \begin{align} &\qquad\qquad\qquad\qquad\qquad M_{12} = \left (kU_1 + lV_1 - c\kappa \right )F_1, \end{align}

(2.10c) \begin{align} &\qquad\qquad\qquad\qquad\qquad M_{21} = \left (kU_2 + lV_2 - c\kappa \right )F_2, \end{align}

(2.10d) \begin{align} & M_{22} = c\kappa \bigl(\kappa ^2 + F_2\bigr) -\kappa ^2 \left (kU_2+lV_2\right ) - F_2 \left (kU_1+lV_1\right ) + k\beta + kB_y - lB_x + i \mu \kappa ^2. \end{align}

To simplify the analysis, we introduce a wavenumber coordinate system, following the approach of Leng & Bai (Reference Leng and Bai2018). In this coordinate system, the unit vectors $\mathbf{e}_\parallel$ and $\mathbf{e}_\perp$ are parallel and perpendicular to $\mathbf{K}$ , respectively. We can then define the following projections:

(2.11a) \begin{align} &\qquad\quad \tilde {U}_i = \frac {\mathbf{K} \boldsymbol{\cdot} \boldsymbol{U}_i}{\kappa } = \frac {kU_i+lV_i}{\kappa } = U_i\cos \theta + V_i\sin \theta , \end{align}

(2.11b) \begin{align} &\qquad\quad\qquad\quad \hat {\beta } = \frac {\mathbf{K} \times \boldsymbol{\nabla} f}{\kappa } = \frac {k\beta }{\kappa } = \beta \cos \theta , \end{align}

(2.11c) \begin{align} & \hat {S} = \frac {\mathbf{K} \times \boldsymbol{\nabla} B}{\kappa } = \frac {kB_y - lB_x}{\kappa } = \frac {f_0}{H_2} \left (\alpha _y\cos \theta - \alpha _x\sin \theta \right ) = \frac {f_0}{H_2}\hat {\alpha }, \end{align}

where $\tilde {U}_i$ is the projection of $\boldsymbol{U}_i$ on $\mathbf{e}_\parallel$ , and $\hat {\beta }$ , $\hat {S}$ and $\hat {\alpha }$ are the projections of $\boldsymbol{\nabla} f$ , $\boldsymbol{\nabla} B$ and $\boldsymbol{\alpha }$ on $\mathbf{e}_\perp$ , respectively. Thus the projections of the PV gradients on $\mathbf{e}_\perp$ are

(2.12a) \begin{align} &\quad\widehat {\boldsymbol{\nabla} Q_1} = F_1\,\Delta \tilde {U} + \hat {\beta }, \end{align}

(2.12b) \begin{align} &\widehat {\boldsymbol{\nabla} Q_2} = -F_2\,\Delta \tilde {U} + \hat {\beta } + \hat {S}, \end{align}

where $\Delta \tilde {U}=\tilde {U}_1 - \tilde {U}_2$ . The projections (2.11) simplify the matrix equation (2.9) to

(2.13) \begin{align} &\begin{pmatrix} c \bigl(\kappa ^2 + F_1 \bigr) - \tilde {U}_1\kappa ^2 - F_1\tilde {U}_2 + \hat {\beta } & \bigl(\tilde {U}_1 - c \bigr)F_1 \\ \bigl(\tilde {U}_2 - c \bigr)F_2 & c\left (\kappa ^2 + F_2\right ) - \tilde {U}_2\kappa ^2 - F_2\tilde {U}_1 + \hat {\beta } + \hat {S} + i \mu \kappa \end{pmatrix} \begin{pmatrix} \hat {\psi }_1 \\ \hat {\psi }_2 \end{pmatrix}\notag\\&\quad = 0. \end{align}

For non-trivial solutions, the determinant of the matrix in (2.13) must be zero. This gives a quadratic equation for $c$ , $A_1 c^2 + A_2 c + A_3 = 0$ , which has the following solution:

(2.14a) \begin{align} &\qquad\qquad\qquad c = \frac {-A_2 \pm \sqrt {A_2^2-4A_1A_3}}{2A_1} \equiv \frac {-A_2 \pm \sqrt {D}}{2A_1},\\[-8pt]\nonumber \end{align}

(2.14b) \begin{align} &\qquad\qquad\qquad\qquad\quad A_1 = \kappa ^2 \bigl(\kappa ^2 + F_1 + F_2 \bigr),\\[-8pt]\nonumber \end{align}

(2.14c) \begin{align} & A_2 = -\kappa ^4\bigl(\tilde {U}_1 + \tilde {U}_2\bigr) - 2\kappa ^2\bigl(F_2\tilde {U}_1 +F_1\tilde {U}_2\bigr) + \hat {\beta } \bigl(2\kappa ^2 + F_1 + F_2 \bigr)\notag\\&\quad + \bigl(\kappa ^2 + F_1 \bigr) \bigl(\hat {S} + i \mu \kappa \bigr),\\[-8pt]\nonumber \end{align}

(2.14d) \begin{align} &A_3 = \kappa ^4 \tilde {U}_1\tilde {U}_2 + \kappa ^2 \left (F_2\tilde {U}_1^2 +F_1\tilde {U}_2^2\right ) - \hat {\beta } \biggl(\bigl(\kappa ^2 + F_1 \bigr) \tilde {U}_2 \notag\\& \quad +\bigl(\kappa ^2 + F_2 \bigr) \tilde {U}_1 - \hat {\beta } - \hat {S} - i \mu \kappa \biggr) - \bigl(\hat {S} + i \mu \kappa \bigr) \Bigl(\kappa ^2 \tilde {U}_1 + F_1 \tilde {U}_2 \Bigr). \end{align}

The solution for $c$ can have both a real part and an imaginary part. The real part, $c_r$ , denotes the eddy phase speed. The imaginary part, $c_i$ , multiplied by the wavenumber magnitude, denotes the unstable growth rate, $\sigma _i = \kappa c_i$ . The requirement for baroclinic instability is that $\sigma _i \gt 0$ .

3.1. Instability conditions in inviscid case

We can use the dispersion relation (2.14) to study the instability of the two-layer model. We start by considering the inviscid case ( $\mu = 0$ ). In this case, a necessary and sufficient condition for instability is that $D = A_2^2 - 4A_1A_3$ in (2.14a ) is negative (as $A_1,A_2,A_3$ are all real, $\sqrt {D}$ is the only possible source of an imaginary part of $c$ ). Here, $D$ is a polynomial in $\kappa$ , and from $D\lt 0$ it follows that there exist both a long-wave and a short-wave cut-off for instability. To get insight into the role of the bottom slope, we derive another necessary condition for instability. We multiply the first row of (2.13) by $d_1 \hat {\kern-1pt \psi}_{1}^\ast / (c-\tilde {U}_1 )$ and the second row by $d_2\hat {\kern-1pt \psi}_{2}^\ast / (c-\tilde {U}_2 )$ , where $d_i \equiv H_i/H$ , and $\ast$ denotes the complex conjugate. Next, we sum the two rows and find

(3.1) \begin{align} &\kappa ^2 \left (d_1 \hat {\psi }_1^2 + d_2 \hat {\psi }_2^2 \right ) + F \bigl(\hat {\psi }_1 - \hat {\psi }_2 \bigr)^2 + \frac {d_1 \hat {\psi }_1^2}{c-\tilde {U}_1} \bigl(F_1\, \Delta \tilde {U} + \hat {\beta } \bigr)\notag\\&\quad + \frac {d_2 \hat {\psi }_2^2}{c-\tilde {U}_2} \bigr(-F_2\, \Delta \tilde {U} + \hat {\beta } + \hat {S} \bigl) = 0, \end{align}

where $F \equiv f_0^2/ (g'H )$ . Both the real and imaginary part of (3.1) must vanish separately. The imaginary part of (3.1) is

(3.2) \begin{align} -c_i \left \lbrace \frac {d_1 \hat {\psi }_1^2}{|c-\tilde {U}_1|^2} \bigl(F_1\, \Delta \tilde {U} + \hat {\beta } \bigr) + \frac {d_2 \hat {\psi }_2^2}{|c-\tilde {U}_2|^2} \bigl(-F_2\, \Delta \tilde {U} + \hat {\beta } + \hat {S} \bigr) \right \rbrace = 0. \end{align}

For $c_i \neq 0$ , which is necessary for instability, (3.2) implies that

(3.3) \begin{align} \bigl( F_1\, \Delta \tilde {U} + \hat {\beta } \bigr) \bigl( -F_2\,\Delta \tilde {U} + \hat {\beta } + \hat {S} \bigr) \lt 0. \end{align}

Note that (3.3) is equivalent to $\widehat {\boldsymbol{\nabla} Q_1}\, \widehat {\boldsymbol{\nabla} Q_2} \lt 0$ . In other words, the component of the PV gradient perpendicular to the wavenumber vector must change sign between the layers for the flow to be unstable. This is the generalisation of the Charney–Stern–Pedlosky criterion (Charney & Stern Reference Charney and Stern1962; Pedlosky Reference Pedlosky1963, Reference Pedlosky1964). A number of interesting aspects emerge from the instability condition (3.3). First, the relevant parameter for stability is the vertical velocity shear $\Delta \tilde {U}$ rather than the individual layer velocities. Also, planetary PV has a stabilising effect since sufficiently large $\hat {\beta }$ will make both $\widehat {\boldsymbol{\nabla} Q_1}$ and $\widehat {\boldsymbol{\nabla} Q_2}$ positive. Note that for meridional wavenumber vectors ( $\theta = \unicode{x03C0} /2$ or $3\unicode{x03C0} /2$ ), planetary PV cannot stabilise the flow, as $\hat {\beta } = 0$ then. Finally, it follows from (3.3) that there exists a critical slope for instability:

(3.4) \begin{align} \hat {\alpha }_c = \frac {f_0\, \Delta \tilde {U}}{g'} - \frac {H_2\hat {\beta }}{f_0}, \end{align}

with the necessary instability conditions

(3.5) \begin{align} \hat {\alpha } &\lt \hat {\alpha }_c \quad \text{if } \widehat {\boldsymbol{\nabla} Q_1} \gt 0,\notag\\ \hat {\alpha } &\gt \hat {\alpha }_c \quad \text{if } \widehat {\boldsymbol{\nabla} Q_1} \lt 0. \end{align}

Thus if the bottom slope component perpendicular to $\mathbf{K}$ exceeds the critical slope, then there is no instability for wavenumber vectors with wave angle $\theta$ . (Here, ‘exceeds’ can mean either to be smaller or greater, depending on the sign of $\widehat {\boldsymbol{\nabla} Q_1}$ .) The term $f_0\,\Delta \tilde {U}/g'$ on the right-hand side of (3.4) represents the slope of the density interface between the two layers, projected on $\mathbf{e}_\parallel$ . This means that on the $f$ -plane, the instability condition (3.5) is that the bottom slope component perpendicular to $\mathbf{K}$ is less steep than the isopycnal slope component parallel to $\mathbf{K}$ . Again, this is a generalisation of a well-known result for zonal flows (e.g. Blumsack & Gierasch Reference Blumsack and Gierasch1972; Mechoso Reference Mechoso1980; Pavec et al. Reference Pavec, Carton and Swaters2005; Isachsen Reference Isachsen2011; Pennel & Kamenkovich Reference Pennel and Kamenkovich2014). Note that (3.5) is not a sufficient condition for instability; as mentioned above, for slopes below the critical slope, there is still instability only for a limited range of wavenumber magnitudes.

3.2. Instability condition with bottom friction

If $\mu \neq 0$ , then the situation changes, as both $A_2$ and $A_3$ in (2.14) have an imaginary part. As seen in the dispersion relation (2.14), including bottom friction is equivalent to adding an imaginary part to the slope parameter $\hat S$ . The imaginary part of $c$ is now given by

(3.6) \begin{align} c_i = \frac {{\textrm {Im}}(-A_2) \pm {\textrm {Im}}(\sqrt {D})}{2A_1}. \end{align}

The necessary and sufficient condition for instability is that (3.6) is positive. The imaginary part of $-A_2$ is equal to $- (\kappa ^2 + F_1 )\mu \kappa$ . To get an expression for the imaginary part of $\sqrt {D}$ , we first write $D$ in polar coordinates:

(3.7) \begin{align} D = r\,{\textrm e}^{{i}\gamma } \implies \sqrt {D} = \sqrt {r}\,{\textrm e}^{{i}\left (\gamma /2 + n\unicode{x03C0} \right )}, \quad n=0,1. \end{align}

The $n=0$ solution of (3.7) corresponds to a positive ${\textrm {Im}} (\sqrt {D} )$ , while the $n=1$ solution yields a negative value. Thus instability implies the sum in (3.6) with $n=0$ , and the difference with $n=1$ . In either case, the condition for instability reduces to

(3.8) \begin{align} \sqrt {r} \sin \left (\gamma /2\right ) \gt \bigl(\kappa ^2 + F_1 \bigr)\mu \kappa . \end{align}

We use the half-angle formula to rewrite the instability condition as (see also Swaters Reference Swaters2009, Reference Swaters2010)

(3.9) \begin{align} 2r \sin ^2(\gamma /2) = r\left (1-\cos \gamma \right ) \gt 2\bigl(\kappa ^2 + F_1 \bigr)^2\mu ^2\kappa ^2, \end{align}

or equivalently,

(3.10) \begin{align} \sqrt {{\textrm {Re}}(D)^2+{\textrm {Im}}(D)^2} \gt {\textrm {Re}}(D) + 2\bigl(\kappa ^2 + F_1 \bigr)^2\mu ^2\kappa ^2. \end{align}

Squaring both sides yields

(3.11) \begin{align} {\textrm {Im}}(D)^2 \gt 4\bigl(\kappa ^2 + F_1 \bigr)^2\mu ^2\kappa ^2\,{\textrm {Re}}(D) + 4\bigl(\kappa ^2 + F_1 \bigr)^4\mu ^4\kappa ^4. \end{align}

We obtain ${\textrm {Re}}(D)$ and ${\textrm {Im}}(D)$ from $D = A_2^2-4A_1A_3$ following (2.14). Some tedious but straightforward algebra yields the following expressions:

(3.12a) \begin{align} {\textrm {Re}}(D) & = \kappa ^4 \bigl(\tilde {U}_1 - \tilde {U}_2 \bigr)^2 \bigl(\kappa ^4 - 4F_1F_2 \bigr) + \bigl(\kappa ^2+F_1\bigr)^2 \Bigl(\hat {S}^2 - \mu ^2\kappa ^2 \Bigr) \notag\\&\quad + 2\hat {\beta }\hat {S} \bigl(\kappa ^2 \left (F_1 - F_2 \right ) + F_1 (F_1 + F_2 ) \bigr) + 2\kappa ^2\hat {S} \bigl(\tilde {U}_1 - \tilde {U}_2 \bigr) \Bigl(\kappa ^4 + F_1\kappa ^2 - 2F_1F_2 \Bigr) \notag\\&\quad + 2\kappa ^4\hat {\beta } \bigl(\tilde {U}_1 - \tilde {U}_2 \bigr) \left (F_1-F_2\right ) + \hat {\beta }^2 (F_1 + F_2 )^2, \end{align}

(3.12b) \begin{align} {\textrm {Im}}(D) & = 2\hat {S} \mu \kappa \bigl(\kappa ^2 + F_1 \bigr)^2 + 2\mu \kappa ^3 \bigl(\tilde {U}_1 - \tilde {U}_2 \bigr)\Bigl(\kappa ^4 + F_1\kappa ^2 - 2F_1F_2 \Bigr) \notag\\&\quad + 2\hat {\beta }\mu \kappa \bigl(\kappa ^2 \left (F_1 - F_2 \right ) + F_1 (F_1 + F_2 ) \bigr). \end{align}

Again, only the vertical velocity shear $\Delta \tilde {U} = \tilde {U}_1 - \tilde {U}_2$ enters the expression. Substituting (3.12) in (3.11) leads eventually to many terms cancelling out. Most notably, all terms containing the bottom slope term $\hat {S}$ disappear. Thus the slope can no longer stabilise the flow in the presence of a bottom Ekman layer. Finally, (3.11) reduces to

(3.13) \begin{align} 16\kappa ^4 F_1 F_2 \mu ^2 \bigl(\kappa ^2 + F_1 + F_2\bigr) \bigl(\Delta \tilde {U}\,\kappa ^2-\hat {\beta }\bigr) \bigl(F_1\, \Delta \tilde {U}+\hat {\beta } \bigr) \gt 0. \end{align}

The first portion is always positive, so for the $\mu \neq 0$ case, there is instability if and only if

(3.14) \begin{align} \bigl(\Delta \tilde {U}\,\kappa ^2-\hat {\beta }\bigr) \bigl(F_1\, \Delta \tilde {U}+\hat {\beta } \bigr) \gt 0. \end{align}

Thus there can be instability only if $\Delta \tilde {U}$ is non-zero. A further constraint that is necessary and sufficient for instability follows from (3.14), depending on the sign of $\Delta \tilde {U} \boldsymbol{\cdot} \hat {\beta }$ :

(3.15a) \begin{align} &\text{for } \Delta \tilde {U} \boldsymbol{\cdot} \hat {\beta } \geqslant 0, \quad |\Delta \tilde {U}| \gt \frac {|\hat {\beta }|}{\kappa ^2}, \end{align}

(3.15b) \begin{align} &\text{for } \Delta \tilde {U} \boldsymbol{\cdot} \hat {\beta } \leqslant 0, \quad |\Delta \tilde {U}| \gt \frac {|\hat {\beta }|}{F_1}. \end{align}

So there is a critical shear for instability; as long as $\Delta \tilde {U}$ is greater than this critical shear, the system is unstable for all wave orientations $\theta$ . If $\Delta \tilde {U}$ and $\hat {\beta }$ have equal signs, then the critical shear is determined by the planetary PV gradient and the wavenumber, and long waves are stable. If $\Delta \tilde {U}$ and $\hat {\beta }$ have opposite signs, then the critical shear depends on the planetary PV gradient and the upper layer deformation radius, and there is no constraint for the wavenumber magnitude. The instability is thus no longer confined to a wavenumber range with both a long-wave and short-wave cut-off, as in the inviscid case; friction destabilises the system (e.g. Holopainen Reference Holopainen1961). Planetary PV can stabilise the flow; on an $f$ -plane, or for meridional wave vectors ( $\hat {\beta }=0$ ), there is instability for all non-zero $\Delta \tilde {U}$ at wave angle $\theta$ . Notably, neither the friction coefficient $\mu$ nor the bottom slope $\hat {\alpha }$ appear in the instability condition (3.14). Frictional destabilisation happens even for infinitesimally small frictional strength, consistent with Swaters (Reference Swaters2009, Reference Swaters2010). Moreover, as long as (3.14) holds, the flow is unstable for all slopes, irrespective of magnitude or orientation. Interestingly, (3.14) also holds for unstable flows in the inviscid case (this follows from combining (3.3) with $A_3\gt 0$ , which must be true for $D\lt 0$ ). However, (3.14) is not sufficient for instability in an inviscid system; constraints for the wavenumber magnitude and slope must be met. By contrast, for $\mu \neq 0$ , (3.14) is necessary and sufficient for instability. The inviscid case is thus a singular limit of the two-layer model.

Figure 1.

Growth rate of unstable waves as a function of wavenumber (normalised by the deformation wavenumber) for different frictional time scales, with parameters as in (3.16). Note the different colour bar ranges. The black curve in each plot shows the wavenumber that maximises the growth rate as a function of the bottom slope.

The impact of bottom slope and bottom friction on baroclinic instability is visualised in figure 1, which shows the growth rates of unstable waves as a function of wavenumber. For this figure, we consider a zonal mean shear and a meridional bottom slope, so that the planetary, topographic and stretching PV are all aligned, even without the transformation to wavenumber coordinates. We consider a zonal wavenumber vector ( $\theta = 0$ ) as this always yields the maximum growth rate. The following dimensional parameters are used, which are representative values for the ocean (e.g. Dohan & Maximenko Reference Dohan and Maximenko2010; Koltermann et al. Reference Koltermann, Gouretski and Jancke2011; LaCasce & Groeskamp Reference LaCasce and Groeskamp2020):

(3.16) \begin{align} &\qquad\qquad f_0 = 10^{-4} \ \textrm{s}^{-1}, \quad \beta = 10^{-11} \ \textrm{m}^{-1}\ \textrm{s}^{-1}, \quad \Delta U = 0.04 \ \textrm{m s}^{-1},\notag \\ &H_1 = 1000 \ \textrm{m}, \quad H_2 = 4000 \ \textrm{m},\quad \rho _1 = 1027.5 \ \textrm{kg m}^{-3},\quad \rho _2 = 1028 \ \textrm{kg m}^{-3}. \end{align}

In this configuration, the deformation radius $L_d = 1/\kappa _d = 1/\sqrt {F_1+F_2}$ is 20 km, and the upper layer deformation radius $L_{d1} = 1/\sqrt {F_1}$ is 22 km. As estimates of the inverse frictional time scale $\mu ^{-1}$ in the ocean are of the order of 1–100 day $^{-1}$ (Arbic & Flierl Reference Arbic and Flierl2004a ), we test different orders of magnitude of $\mu$ . Furthermore, we test bottom slope magnitudes of the order of $10^{-4}{-}10^{-3}$ , which capture most of the open ocean (LaCasce Reference LaCasce2017). As we consider an eastward mean shear in the Northern Hemisphere, positive slopes (rising towards the north) are retrograde, and negative slopes are prograde in this configuration. Figure 1(a) demonstrates the existence of a critical slope for instability in the inviscid case, resulting in a strong asymmetry between retrograde and prograde slopes (Blumsack & Gierasch Reference Blumsack and Gierasch1972; Mechoso Reference Mechoso1980). Figure 1(b) shows that even weak bottom friction destabilises the system for all bottom slopes, and removes the short-wave cut-offs (as $\Delta U \boldsymbol{\cdot} \beta \gt 0$ here, there is still a long-wave cut-off at $\kappa = \sqrt {\beta /\Delta U}$ ; see (3.15)), but also that the growth rates become weaker. There is still strong asymmetry between retrograde and prograde slopes: for retrograde slopes, growth rates are much weaker, and the maximum growth occurs at smaller wavenumbers than for prograde slopes. With increasing frictional strength (figure 1 c,d), the growth rates decrease further, and the asymmetry between retrograde and prograde slopes gets weaker. Thus there is a shift from a slope-dominated regime to a friction-dominated regime. This can be understood from (2.14) by noting that the terms representing the topographic PV gradient and the bottom friction always appear together as the sum $\hat {S} + i \mu \kappa$ ; as $\mu$ increases, so does its relative importance over the topographic term.

Figure 2 shows the maximum growth rate as a function of bottom slope, for both positive and negative zonal shear configurations. For no or weak friction, the most unstable growth rate depends strongly and asymmetrically on the bottom slope. The growth rates are much higher for prograde slopes than for retrograde slopes in this parameter configuration. For stronger, but realistic friction, the growth rate curves flatten, illustrating the shift from the slope-dominated regime to the friction-dominated regime. The growth rates are very low for strong friction. Hence even though the system is unstable in an analytical sense, the unstable modes grow only very slowly for strong friction.

Figure 2.

The maximum growth rate as a function of the bottom slope for different frictional strengths, for positive zonal shear $\Delta U = 0.04 \ {\textrm{m s}}^{-1}$ and negative zonal shear $\Delta U = -0.04 \ {\textrm{m s}}^{-1}$ , and the other model parameters as in (3.16).

3.3. Instability characteristics for varying orientations of the PV gradients

We now consider how the instability characteristics of the two-layer model change for varying orientations of the mean shear and the topography. For simplicity, we set $\boldsymbol{U}_2 = 0$ so that the shear vector is $\boldsymbol{U}_1 \equiv \boldsymbol{U}$ . Figure 3 shows the most unstable growth rate as a function of the bottom slope vector, for shear vectors making angles $45^\circ$ (top row) or $300^\circ$ (bottom row) with the zonal direction. In these plots, the distance to the origin is the magnitude (steepness) of the bottom slope, and the angle around the $x$ -axis is the direction of the slope (i.e. the direction in which the water column becomes shallower). For each slope vector, growth rates are computed for a range of wavenumber vectors (varying magnitudes $\kappa$ and orientations $\theta$ ), and the maximum growth rate is plotted. Shear orientations other than $45^\circ$ or $300^\circ$ show similar results (not shown here). Figures 3(a) and 3(d) show that in the inviscid case, the system is stable (white region where gridlines are visible) only for a very narrow range of slope vectors –namely, close to the slope for which the lower layer PV gradient $\boldsymbol{\nabla} Q_2$ in (2.4b ) is perpendicular to $\boldsymbol{U}$ . Note that $\boldsymbol{\nabla} Q_2 \perp \boldsymbol{U}$ is equivalent to the planetary, topographic and stretching PV gradients all being aligned with each other. The red lines in the figures indicate the slope magnitudes and orientations for which $\boldsymbol{\nabla} Q_2 \perp \boldsymbol{U}$ . This line and the stability region do not start at the origin: only for sufficiently steep slopes can the system become stable. For both $45^\circ$ and $300^\circ$ shear angles, stability occurs for retrograde slopes, i.e. seafloors deepening towards the right of the mean shear (in the Northern Hemisphere); prograde slopes, on the other hand, are always unstable. Moreover, as soon as $\boldsymbol{\nabla} Q_2$ is no longer perpendicular to $\boldsymbol{U}$ , the system becomes unstable. This does not necessarily mean that the system is unstable for all wavenumber vectors – for example, it will be stable for wavenumber magnitudes outside the long-wave/short-wave cut-offs, and for wavenumber orientations for which (3.3) does not hold (see e.g. figure 4 in Leng & Bai Reference Leng and Bai2018). However, figures 3(a) and 3(d) show that for $\boldsymbol{\nabla} Q_2$ not perpendicular to $\boldsymbol{U}$ , there is always at least some wavenumber vector for which $\sigma _i$ is positive. As seen before, figure 3(b,c,e,f) demonstrate that the presence of bottom friction destabilises the system for all bottom slopes, but also makes the growth rates (much) weaker. Figure 3(a)–3(f) all show that the growth rates are symmetric around the line $\boldsymbol{\nabla} Q_2 \perp \boldsymbol{U}$ . Growth rates increase as the slope becomes more aligned with the mean shear (so the topographic PV gradient becomes more perpendicular to the stretching PV gradient), and are generally higher for prograde slopes. As friction increases, the dependency of the growth rate on the slope weakens, and with that, the asymmetry between prograde and retrograde slopes: the growth rates become very weak for all slope orientations and magnitudes.

Figure 3.

The most unstable growth rate as functions of the bottom slope vector for different orientations of the mean shear and different frictional strengths. The axes indicate the magnitude and orientation of the slope. The direction of the shear vector is indicated by the arrow in each plot, and the shear magnitude is 0.04 m s^–1; the other model parameters are as in (3.16). The black dot marks the origin of the plot. The red lines in (a), (b), (d) and (e) indicate the orientation of the slope at which the lower layer PV gradient is perpendicular to the shear, as a function of the slope magnitude.

Figure 4.

As figure 3 but showing the most unstable wavenumber. The $\kappa = \kappa _d$ contour is indicated in white.

Figure 5.

As figure 3 but showing the propagation direction of the most unstable wave.

Figure 4 shows the most unstable wavenumber as a function of bottom slope orientation and magnitude. Generally, retrograde slopes favour lower wavenumbers (larger scales), whereas prograde slopes favour higher wavenumbers (smaller scales), as in Leng & Bai (Reference Leng and Bai2018). An interesting feature occurs around the line $\boldsymbol{\nabla} Q_2 \perp \boldsymbol{U}$ in the inviscid and weak friction cases: there is a discontinuity across this line with a switch from a low wavenumber mode to a high wavenumber mode. The discontinuity goes in opposite directions for the two shear orientations considered here; the shift from low to high wavenumber occurs in the anticlockwise direction for shear angle $45^\circ$ , and in the clockwise direction for shear angle $300^\circ$ . Other shear orientations between $0^\circ$ and $270^\circ$ show the same behaviour as for $45^\circ$ , and other shear orientations between $270^\circ$ and $360^\circ$ show the same as for $300^\circ$ (not shown here). The discontinuity across $\boldsymbol{\nabla} Q_2 \perp \boldsymbol{U}$ is smoothed for strong friction, and the most unstable wavenumber becomes more uniform for all bottom slope magnitudes and orientations. However, the most unstable wavenumber remains asymmetric in the line $\boldsymbol{\nabla} Q_2 \perp \boldsymbol{U}$ for retrograde slopes, as opposed to the maximum growth rate. Note that periodic patterns appear, most notably in the low wavenumber mode close to the line $\boldsymbol{\nabla} Q_2 \perp \boldsymbol{U}$ . These are in part due to the resolution of the values of $\kappa$ and $\theta$ that were tested, but some periodic signal still remains even for higher resolution (not shown here). This is likely due to interactions of higher harmonics. As the patterns appear within a regime where the growth rates are very weak, they are not of great importance for the qualitative behaviour of the instability as a function of slope.

Finally, figure 5 shows the propagation direction of the most unstable wave as a function of the bottom slope. If the phase speed (real part of $c$ ) of the most unstable wave is positive, then the propagation direction is given by the orientation $\theta$ of the most unstable wavenumber vector; if the phase speed is negative, then the propagation direction is exactly opposite to $\theta$ (shift of $180^\circ$ ). For the inviscid and weak friction cases, again there is a clear asymmetry between prograde and retrograde slopes. For prograde slopes, the propagation direction is close to the direction of the mean shear. For retrograde slopes, on the other hand, the most unstable wave crosses the mean flow, until the propagation direction is close to the normal direction of the mean shear around the line $\boldsymbol{\nabla} Q_2 \perp \boldsymbol{U}$ . This is in agreement with the case studies considered by Leng & Bai (Reference Leng and Bai2018). As in figure 4, a discontinuity occurs across $\boldsymbol{\nabla} Q_2 \perp \boldsymbol{U}$ : the propagation direction switches by $180^\circ$ across this line. This switch ensures that the most unstable wave always propagates at an acute angle to the mean shear. Over a retrograde slope with $\boldsymbol{\nabla} Q_2 \boldsymbol{\cdot} \boldsymbol{U} \gt 0$ , the most unstable wave travels upslope, with the mean shear to its right; if $\boldsymbol{\nabla} Q_2 \boldsymbol{\cdot} \boldsymbol{U} \lt 0$ , then it travels downslope with $\boldsymbol{U}$ to its left. For strong friction, the discontinuity across $\boldsymbol{\nabla} Q_2 \perp \boldsymbol{U}$ disappears, as does the prograde–retrograde asymmetry: the waves all travel parallel to the mean shear.