Rayleigh–Kuo criterion

The presence of these instabilities in a rotating fluid have been observed in laboratory experiments. The settings of the experiment were based on the conditions in the Gulf Stream and showed that within the ocean currents such as the Gulf Stream, it is possible for barotropic instabilities to occur.[6] But barotropic instabilities were also observed in other Western Boundary Currents (WBC). In the Agulhas current, the barotropic instability leads to ring shedding. The Agulhas current retroflects (turns back) near the coast of South Africa. At this same location, some anti-cyclonic rings of warm water escape from the mean current and travel along the coast of Africa. The formation of these rings is a manifestation of a barotropic instability.[7]

In order to derive the Rayleigh–Kuo criterion, some assumptions are made on the fluids properties. We consider a nondivergent, two-dimensional barotropic fluid. The fluid has a mean zonal flow direction which can vary in the meridional direction. On this mean flow, some small perturbations are imposed in both the zonal and meridional direction: ${\displaystyle u(y,t)=U(y)+u^{*}(y,t)}$ and ${\displaystyle v=v^{*}}$. The perturbations need to be small in order to linearize the vorticity equation. Vertical motion and divergence and convergence of the fluid are neglected. When taking into account these factors, a similar result would have been obtained with only a small shift in the position of the criterion within the velocity profile.[1]

The derivation of the Kuo criterion will be done within the domain ${\displaystyle L=[0,y]}$. On the northern and southern boundary of this domain, the meridional fluid is zero.

To derive the Rayleigh equation for a barotropic fluid, the barotropic vorticity equation is used. This equation assumes that the absolute vorticity is conserved: ${\displaystyle {\frac {d\zeta _{a}}{dt}}=0}$ here, ${\displaystyle {\frac {d}{dt}}}$ is the material derivative. The absolute vorticity is the relative vorticity plus the planetary vorticity: ${\displaystyle \zeta _{a}=\zeta +f}$. The relative vorticity, ${\displaystyle \zeta }$, is the rotation of the fluid with respect to the Earth. The planetary vorticity (also called Coriolis frequency),${\displaystyle f}$, is the vorticity of a parcel induced by the rotation of the Earth. When applying the beta-plane approximation for the planetary vorticity, the conservation of absolute vorticity looks like:

$${\displaystyle {\frac {d\zeta _{a}}{dt}}={\frac {d}{dt}}\left(\zeta +\beta y\right)=0}$$

The relative vorticity is defined as ${\displaystyle \zeta ={\frac {\partial v}{\partial x}}-{\frac {\partial u}{\partial y}}.}$ Since the flow field consist of a mean flow with small perturbations, it can be written as ${\displaystyle \zeta ={\overline {\zeta }}+\zeta ^{*}}$ with ${\displaystyle {\overline {\zeta }}=-{\frac {\partial U}{\partial y}}}$ and ${\displaystyle \zeta ^{*}={\frac {\partial v^{*}}{\partial x}}-{\frac {\partial u^{*}}{\partial y}}.}$ This formulation is used in the vorticity equation:

$${\displaystyle {\begin{aligned}0&={\frac {d}{dt}}\left(\zeta +\beta y\right)\\0&={\frac {d}{dt}}\left(\zeta '+{\overline {\zeta }}+\beta y\right)\\0&=\left({\frac {\partial }{\partial t}}+u{\frac {\partial }{\partial x}}+v{\frac {\partial }{\partial y}}\right)\left(\zeta ^{*}-{\frac {\partial U}{\partial y}}+\beta y\right)\end{aligned}}}$$

Here, ${\displaystyle u}$ and ${\displaystyle v}$ are the zonal and meridional components of the flow and ${\displaystyle \zeta '}$ is the relative vorticity induced by the perturbations on the flow (${\displaystyle u'}$ and ${\displaystyle v'}$). ${\displaystyle U}$ is the mean zonal flow and ${\displaystyle \beta }$ is derivative of the planetary vorticity ${\displaystyle f}$ with respect to ${\displaystyle y}$ .

A zonal mean flow with small perturbations was assumed, ${\displaystyle u=U+u^{*}}$, and a meridional flow with a zero mean, ${\displaystyle v=v^{*}}$. Since it was assumed that the perturbations are small, a linearization can be performed on the barotropic vorticity equation above, ignoring all the non-linear terms (terms where two or more small variables, i.e. ${\displaystyle u^{*},v^{*},\zeta ^{*}}$, are multiplied with one another). Also the derivative of ${\displaystyle u}$ in the zonal direction, the time derivative of the mean flow ${\displaystyle U}$ and the time derivative of ${\displaystyle \beta y}$ are zero. This results in a simplified equation:

$${\displaystyle {\begin{aligned}0&=\left({\frac {\partial }{\partial t}}+U{\frac {\partial }{\partial x}}\right)\zeta '-v'{\frac {\partial }{\partial y}}{\frac {\partial U}{\partial y}}+v'{\frac {\partial }{\partial y}}\left(\beta y\right)\\0&=\left({\frac {\partial }{\partial t}}+U{\frac {\partial }{\partial x}}\right)\zeta '+v'\left(\beta -{\frac {\partial ^{2}U}{\partial y^{2}}}\right).\end{aligned}}}$$

With ${\displaystyle \zeta '}$ as defined above (${\displaystyle \zeta ^{*}={\frac {\partial v^{*}}{\partial x}}-{\frac {\partial u^{*}}{\partial y}}}$) and ${\displaystyle u^{*}}$ and ${\displaystyle v^{*}}$ the small perturbations in the zonal and meridional components of the flow.

To find the solution to the linearized equation, a stream function was introduced by Lord Rayleigh for the perturbations of the flow velocity:

$${\displaystyle u^{*}={\frac {\partial \psi }{\partial y}},\;\;\;\;v^{*}={\frac {-\partial \psi }{\partial x}}.}$$

These new definitions of the stream function are used to rewrite the linearized barotropic vorticity equation.

$${\displaystyle {\begin{aligned}0&=\left({\frac {\partial }{\partial t}}+U{\frac {\partial }{\partial x}}\right)\left({\frac {\partial v^{*}}{\partial x}}-{\frac {\partial u^{*}}{\partial y}}\right)+v^{*}\left(\beta -{\frac {\partial ^{2}U}{\partial y^{2}}}\right)\\0&=\left({\frac {\partial }{\partial t}}+U{\frac {\partial }{\partial x}}\right)\left(-{\frac {\partial ^{2}\psi }{\partial x^{2}}}-{\frac {\partial ^{2}\psi }{\partial y^{2}}}\right)-{\frac {\partial \psi }{\partial x}}\left(\beta -U''\right)\\0&=\left({\frac {\partial }{\partial t}}+U{\frac {\partial }{\partial x}}\right)\nabla ^{2}\psi +{\frac {\partial \psi }{\partial x}}\left(\beta -U''\right)\end{aligned}}}$$

Here, ${\displaystyle U''}$ is the second derivative of ${\displaystyle U}$ with respect to ${\displaystyle y}$ ${\displaystyle (U''={\frac {\partial ^{2}U}{\partial y^{2}}})}$. To solve this equation for the stream function, a wave-like solution was proposed by Rayleigh which reads ${\displaystyle \psi (x,y,t)=\Psi (y)e^{i\alpha \left(x-ct\right)}}$. The amplitude ${\displaystyle \Psi (y)}$ may be complex number, ${\displaystyle \alpha }$ is the wave number which is a real number and ${\displaystyle c}$ is the phase velocity which may be complex as well. Inserting this proposed solution leads us to the equation which is known as Rayleigh's equation.

$${\displaystyle {\begin{aligned}0&=\left({\frac {\partial }{\partial t}}+U{\frac {\partial }{\partial x}}\right)\nabla ^{2}\psi +{\frac {\partial \psi }{\partial x}}\left(\beta -U''\right)\\[14pt]0&=\left({\frac {\partial }{\partial t}}+U{\frac {\partial }{\partial x}}\right)\left((i\alpha )^{2}\Psi e^{i\alpha \left(x-ct\right)}+\Psi ''e^{i\alpha \left(x-ct\right)}\right)+i\alpha \Psi e^{i\alpha (x-ct)}(\beta -U'')\\[14pt]0&=(i\alpha )^{2}\Psi e^{i\alpha (x-ct)}(-i\alpha c+i\alpha U)+\Psi ''e^{i\alpha (x-ct)}(-i\alpha c+i\alpha U)+i\alpha \Psi e^{i\alpha (x-ct)}(\beta -U'')\\[14pt]0&=i\alpha e^{i\alpha (x-ct)}[(\alpha ^{2}c-\alpha ^{2}U)\Psi +\Psi ''(-c+U)+\Psi (\beta -U'')]\\[14pt]0&=(U-c)(\Psi ''-\alpha ^{2}\Psi )+(\beta -U'')\Psi )\\[14pt]\end{aligned}}}$$

To get to this equation, in the last step it was used that ${\displaystyle \alpha }$ can't be zero and neither can the exponential. This means that the terms in the square brackets needs to be zero. The symbol ${\displaystyle \Psi ''}$ denotes the second derivative of the amplitude of the stream function, ${\displaystyle \Psi }$ with respect to ${\displaystyle y}$ ${\displaystyle (\Psi ''={\frac {\partial ^{2}\Psi }{\partial y^{2}}})}$. This last equation that was derived, is known as Rayleigh's equation which is a linear ordinary differential equation. It is very difficult to explicitly solve this equation. It is therefore that Hsiao-Lan Kuo came up with a stability criterion for this problem without actually solving it.

Instead of solving Rayleigh's equation, Hsiao-Lan Kuo came up with a necessary stability condition which had to be met in order for the fluid to be able to get unstable. To get to this criterion, Rayleigh's equation was rewritten and the boundary conditions of the flow field are used.

The first step is to divide Rayleigh's equation by ${\displaystyle (U-c)}$ and multiplying the equation by the complex conjugate of ${\displaystyle \Psi \;\;(\Psi ^{*}=\Psi _{r}-i\Psi _{i})}$.

$${\displaystyle {\begin{aligned}0&=(\Psi ''-\alpha ^{2}\Psi )+\left({\frac {\beta -U''}{U-c}}\right)\Psi \\0&=\Psi ^{*}(\Psi _{r}''+i\Psi _{i}''-\alpha ^{2}\Psi _{r}-\alpha ^{2}i\Psi _{i})+\Psi ^{*}\left({\frac {\beta -U''}{U-c}}\right)(\Psi _{r}-i\Psi _{i})\\0&=\Psi _{r}\Psi _{r}''+\Psi _{i}\Psi _{i}''-\alpha ^{2}(\Psi _{r}^{2}-\Psi _{i}^{2})+\left({\frac {\beta -U''}{U-c}}\right)(\Psi _{r}^{2}-\Psi _{i}^{2})+i(-\Psi _{i}\Psi _{r}''+\Psi _{r}\Psi _{i}'')\\0&=\Psi _{r}''\Psi _{r}+\Psi _{i}''\Psi _{i}+\left(-\alpha ^{2}+{\frac {U-c_{r}}{|U-c|^{2}}}(\beta -U'')\right)|\Psi |^{2}+i\left({\frac {c_{i}}{|U-c|^{2}}}(\beta -U'')|\Psi |^{2}-\Psi _{r}''\Psi _{i}+\Psi _{i}''\Psi _{r}\right)\end{aligned}}}$$

In the last step, ${\displaystyle (U-c)}$ is multiplied with its complex conjugate leading to the following equality is used: ${\displaystyle {\frac {1}{U-c}}={\frac {1}{U-c_{r}-ic_{i}}}={\frac {U-c_{r}+ic_{i}}{|U-c|^{2}}}}$. For the solution of Rayleigh's equation to exist, both the real and imaginary part of the equation above need to be equal to zero.

To get to the Kuo criterion, the imaginary part is integrated over the domain (${\displaystyle y=[0,L]}$) . The stream function at the boundaries of the domain is zero, ${\displaystyle \Psi (0)=\Psi (L)=0}$, as already stated in the assumptions. The zonal flow must vanish at the boundaries of the domain. This leads to a constant stream function which is set to zero for convenience.

${\displaystyle \int _{0}^{L}(\Psi _{r}\Psi _{i}''-\Psi _{i}\Psi _{r}'')dy+\int _{0}^{L}\left(c_{i}{\frac {|\Psi |^{2}}{|U-c|}}(\beta -U'')\right)=0}$

The first integral can be solved:

${\displaystyle {\begin{aligned}\int _{0}^{L}(\Psi _{r}\Psi _{i}''-\Psi _{i}\Psi _{r}'')dy&=\int _{0}^{L}{\frac {\partial }{\partial y}}(\Psi _{r}\Psi _{i}'-\Psi _{i}\Psi _{r}')dy\\[8pt]&=(\Psi _{r}\Psi _{i}'-\Psi _{i}\Psi _{r}')|_{0}^{L}\\[8pt]&=0\\[12pt]\end{aligned}}}$

So the first integral is equal to zero. This means that the second integral should also be zero, making it possible to solve this integral numerically.

${\displaystyle {\begin{aligned}\int _{0}^{L}\left(c_{i}{\frac {|\Psi |^{2}}{|U-c|}}(\beta -U'')\right)dy&=0\\\end{aligned}}}$

When ${\displaystyle c_{i}}$ is zero, we are dealing with a stable amplitude of the solution, this means that the solution is stable. We are looking for un unstable situation, so then ${\displaystyle {\frac {|\Psi |^{2}}{|U-c|}}(\beta -U'')}$ should be zero. Since the fraction in front of ${\displaystyle (\beta -U'')}$ is non-zero and positive, this leads to the conclusion that ${\displaystyle (\beta -U'')}$ should be zero. This leads to the final formulation, the Kuo criterion:

$${\displaystyle {\begin{aligned}\beta -U''&=0\\[10pt]\beta &=U''\\[10pt]{\frac {\partial (\beta y)}{\partial y}}&={\frac {\partial ^{2}U}{\partial y^{2}}}\end{aligned}}}$$

Here, ${\displaystyle U}$ is the mean zonal flow and ${\displaystyle \beta }$ is the derivative of the planetary vorticity ${\displaystyle f}$ with respect to ${\displaystyle y}$.