Sloping convection

These baroclinic instabilities may be initiated by the process of 'sloping convection' or 'slanted thermal convection'. To understand this, consider a fluid in steady state and under the thermal-wind balance. Initially, a fluid parcel is at location A. The fluid parcel is slightly perturbed to location B, while still retaining its original density. Therefore, the fluid parcel is now in a location with a lower density than itself and the parcel will just sink down to its original position; the fluid parcel is now stable. However, when a parcel displaced to location C, it is surrounded by fluid with a higher density than the parcel itself. Due to its relatively low density with respect to its surroundings, the parcel will float up even further. Now a small perturbation grows into a larger one, which implies a baroclinic instability.

A criterion for an instability to occur can be defined. As stated before, in a baroclinic fluid, the thermal-wind balance holds, which implies the following two relations:

${\displaystyle {\begin{aligned}{\frac {\partial v}{\partial z}}={\frac {g}{\rho f}}{\frac {\partial \rho }{\partial x}}\end{aligned}}}$ and ${\displaystyle {\begin{aligned}{\frac {\partial u}{\partial z}}=-{\frac {g}{\rho f}}{\frac {\partial \rho }{\partial y}}\end{aligned}}}$,

where ${\displaystyle \rho }$ is the density and ${\displaystyle x}$, ${\displaystyle y}$ and ${\displaystyle z}$ are the spatial coordinates in the horizontal (latitudinal and longitudinal) and vertical direction, respectively. ${\displaystyle u}$ and ${\displaystyle v}$ represent the horizontal (zonal and meridional) components of the velocity vector ${\displaystyle {\mathbf {u}}}$ in the ${\displaystyle x}$- and ${\displaystyle y}$-direction, respectively. Now thus ${\displaystyle {\frac {\partial \rho }{\partial x}}}$ and ${\displaystyle {\frac {\partial \rho }{\partial y}}}$ are the two horizontal density gradients. ${\displaystyle g}$ is the gravitational acceleration at the surface of the Earth and ${\displaystyle f}$ the Coriolis parameter.

Therefore a horizontal density gradient in the ${\displaystyle y}$-direction ${\displaystyle \left({\frac {\partial \rho }{\partial y}}\right)}$ leads to a gradient in horizontal flow velocity ${\displaystyle u}$ over depth ${\displaystyle \left({\frac {\partial u}{\partial z}}\right)}$.

The slope of the displacement is defined as

${\displaystyle {\frac {\Delta z}{\Delta y}}={\frac {w'\Delta t}{v'\Delta t}}={\frac {w'}{v'}}}$,

where ${\displaystyle v'}$ and ${\displaystyle w'}$ are the horizontal and vertical velocities of the perturbation, respectively.

An instability now occurs when the slope of the displacement is smaller than the slope of the isopycnals. The isopycnals can be mathematically described as ${\displaystyle z=H+{\bar {a}}}$. Now this results in an instability when:

${\displaystyle {\begin{aligned}{\frac {d{\bar {a}}}{dy}}>{\frac {w'}{v'}}.\end{aligned}}}$

From now on, only a two layer system with ${\displaystyle U_{1}}$ and ${\displaystyle U_{2}}$ the slopes of the top and bottom layer, respectively, is considered to simplify the problem. This is now similar to the classic Philips model. ^[9] From the thermal-wind balance it now follows that

${\displaystyle {\begin{aligned}{\frac {d{\bar {a}}}{dy}}={\frac {f_{0}}{g'}}\left(U_{1}-U_{2}\right),\end{aligned}}}$

where ${\displaystyle g'={\frac {g(\rho _{2}-\rho _{1})}{\rho _{0}}}}$ is the reduced gravity and ${\displaystyle f_{0}}$ the Coriolis-parameter at the equator according to the beta-plane approximation.

Performing a scale analysis on the slope of the perturbation allows to assign physical quantities to this mathematical problem. This now results in

${\displaystyle {\begin{aligned}{\frac {w'}{v'}}\sim {\frac {H}{L}}{\frac {U_{1}-U_{2}}{f_{0}L}}\sim {\frac {H}{L}}{\frac {\beta _{0}L}{f_{0}}}\end{aligned}}}$,

where ${\displaystyle H}$ is the scale height, ${\displaystyle L}$ the horizontal length scale, and ${\displaystyle \beta _{0}}$ is the Rossby-parameter.

From this it can be stated that an instability occurs when

${\displaystyle {\begin{aligned}{\frac {f_{0}^{2}L^{2}}{g'H}}>1\end{aligned}}}$ or ${\displaystyle {\begin{aligned}{\frac {f_{0}^{2}}{g'H}}{\frac {\Delta U}{\beta _{0}}}>1\end{aligned}}}$,

where ${\displaystyle g'={\frac {g(\rho _{2}-\rho _{1})}{\rho _{0}}}}$ is the reduced gravity and  ${\displaystyle \Delta U=U_{2}-U_{1}}$ is the velocity difference between the lower and upper layer. This criterion can be used to identify whether a small perturbation will grow into a larger one and thus whether an instability is expected to occur. From this it follows that you need some kind of shear ${\displaystyle \Delta U}$ to obtain an instability, it is easier to get an instability for long waves (perturbations) with large ${\displaystyle L}$, and the ${\displaystyle \beta _{0}}$ and therefore the beta-effect is stabilizing. ^[7]

Furthermore, for the baroclinic Rossby radius of deformation it holds that ${\displaystyle R\sim {\frac {\sqrt {g'H}}{f_{0}}}}$. Now the instability criteria simplify to

${\displaystyle {\begin{aligned}L\gtrsim R\end{aligned}}}$ or ${\displaystyle {\begin{aligned}\Delta U\gtrsim \beta _{0}R^{2}\end{aligned}}}$.

From this analysis it also follows that baroclinic instabilities are important for small Rossby numbers, where ${\displaystyle Ro={\frac {U}{Lf}}}$.