Recharge oscillator

The idealised non-dimensional theory that has been proposed by Jin in 1997[1] to explain ENSO consists of the mathematical structure described below. It relies on modelling the western and eastern part of the Pacific Ocean as two pools. The assumptions and equations behind the model are explained below.

The thermocline depth anomaly in the eastern part of the basin ${\displaystyle h_{E}}$ is directly and instantaneously related to the anomaly in the western part ${\displaystyle h_{W}}$ and to the wind stress anomaly ${\displaystyle \tau }$, according to the relation

${\textstyle h_{E}=h_{W}+\tau }$.

The thermocline depth anomaly changes over the western equatorial Pacific are mathematically described by the equation

${\displaystyle {\frac {dh_{W}}{dt}}=-rh_{W}-F_{\tau }}$

where ${\displaystyle -rh_{W}}$ represents the ocean adjustment, characterised by a damping process rate ${\displaystyle r}$ due to the mixing and equatorial energy loss to the boundary layer currents, which occur at the eastern and western side of the basin. The second term ${\textstyle -F_{\tau }}$ represents the Sverdrup transport across the basin, or equally the heat transport into or out of the basin; Sverdrup transport depends on the wind stress curl ${\displaystyle \nabla \times \tau }$. Since ${\displaystyle F_{\tau }}$ is directly proportional to zonally integrated wind stress and its curl, it is possible to approximate ${\textstyle F_{\tau }=\alpha \tau }$, where ${\textstyle \alpha }$ is a constant. The minus sign in the equation above is due to the fact that a westerly wind stress reduces the thermocline depth in the western basin, while an enforced trade wind (whose direction is always east to west) increases the thermocline depth.

The previous equations provide a simplified description of the basinwide equatorial oceanic adjustment under the anomalous wind stress forcing.

The SST anomaly ${\displaystyle T_{E}}$ evolution in time is described by the relation

${\displaystyle {\frac {dT_{E}}{dt}}=-cT_{E}+\gamma h_{E}+\delta _{s}\tau _{E}}$

where ${\textstyle -cT_{E}}$ represents the SST relaxation due to the ${\displaystyle c}$ rate damping processes, ${\textstyle +\gamma h_{E}}$ is taking account of the climatological upwelling, and ${\textstyle +\delta _{s}\tau _{E}}$ represents the advective feedback. ${\textstyle \tau _{E}}$ is the wind stress averaged over the region where the SST occurs, ${\textstyle \gamma }$ and ${\textstyle \delta _{s}}$ are respectively the thermocline and the Ekman pumping feedback coefficients.

As explained in the previous section, the atmospheric response to a SST anomaly is an increased wind stress ${\displaystyle \tau }$, whose orientation depends on anomaly's sign. The wind stress anomaly magnitude is influenced by the zonal area where the SST is averaged, and results bigger if all the basin is taken into account, rather than just its eastern part.[11] This observation allows to approximate the relation between ${\textstyle \tau }$, ${\textstyle \tau _{E}}$ and ${\displaystyle T_{E}}$:

${\displaystyle \tau =bT_{E},\tau _{E}=b'T_{E}}$

with ${\displaystyle b,b'}$ coupling coefficients.

From the previous equations it is possible to derive a linear coupled system, which describes the thermocline depth and the SST anomaly time evolution in the eastern side of the basin:

${\displaystyle {\frac {dh_{W}}{dt}}=-rh_{W}-\alpha bT_{E}}$

${\displaystyle {\frac {dT_{E}}{dt}}=RT_{E}+\gamma h_{W}}$

where ${\displaystyle R}$ is initially defined as the sum of the already introduced constants ${\displaystyle \gamma b,\delta _{s}b'}$ and ${\displaystyle -c}$, and describes the Bjerknes' positive feedback hypothesis. As already stated ${\textstyle \gamma b}$ represents the thermocline feedback, ${\displaystyle \delta _{s}b'}$ the Ekman upwelling, and ${\textstyle -c}$ the rate of the damping process. Because of the weak local wind stress averaged over the eastern basin, the Ekman upwelling feedback parameter ${\displaystyle \delta _{s}b'}$is negligible when compared to the other two terms, ultimately leading to ${\displaystyle R=\gamma b-c}$.

The model presented above is still highly idealised. There is a similar approach which investigates the same climatological anomalies in the Walker circulation along with ocean surface-layer thickness anomalies, but with a more physical perspective.[12]

The key processes of this recharge-oscillator model still involve the ocean (dynamics, volume conservation and heat budget) and the atmosphere, thus leading to another coupled model that comprehensively describes the mechanism of the different phases of ENSO. The assumptions are slightly different from those of the model described above.

The contribution from the ocean is updated to be that of a reduced-gravity surface layer with an average thickness ${\displaystyle H}$. Following a similar approach as described above, given the wind stress anomaly ${\displaystyle \tau }$ and one thermocline depth anomaly (for instance ${\displaystyle h_{W}}$) it is possible to know the other depth anomaly (${\displaystyle h_{E}}$) through the relation:[12]

${\displaystyle g'{\frac {h_{E}-h_{W}}{L}}={\frac {\tau }{\rho _{0}H}}}$

where ${\displaystyle g'}$ is the reduced gravity ${\displaystyle g'=g{\frac {\Delta \rho }{\rho _{0}}}}$. By imposing the volume conservation it is possible to see that the only transport that can exist is a meridional transport. This is accomplished through the Sverdrup transport induced by the wind stress anomaly.

The vertical integrated North-South velocity is then:

${\displaystyle V_{TOT}={\frac {1}{\rho _{0}\beta _{0}}}({\frac {\partial \tau ^{y}}{\partial x}}-{\frac {\partial \tau ^{x}}{\partial y}})}$.

Nevertheless, in this model we can consider no wind stress anomaly in the meridional direction, leading to a final integrated transport:

${\displaystyle V_{TOT}=-{\frac {1}{\rho _{0}\beta _{0}}}{\frac {\partial \tau ^{x}}{\partial y}}}$.

The wind anomaly is limited in space, and decreases away from the equator; its limit are set by the equatorial Rossby radius of deformation ${\displaystyle R_{EQ}}$, that at the equator assumes a value slightly higher than 200 Km. Therefore, the wind anomaly is considered to be maximum at the equator and to reach ${\displaystyle \tau =0}$ over the limits ${\displaystyle \pm R_{EQ}}$ and beyond them. By virtue of this consideration the meridional integrated transport and the corresponding flow divergence ${\displaystyle D}$ can be calculated as:

${\displaystyle V_{TOT}=\pm {\frac {1}{\rho _{0}\beta _{0}}}{\frac {\tau }{R_{EQ}}};\ \ \ \ \ D\sim {\frac {\tau }{\rho _{0}\beta _{0}R_{EQ}^{2}}}}$

where the ${\displaystyle \pm }$ sign in the total flow refers to the northern boundary (${\displaystyle +}$) and the southern boundary (${\displaystyle -}$).

On the basis of what has just been described, the understanding and the inclusion of the ocean dynamics contribution is completed, and it is therefore possible to estimate the total change in the western thermocline depth as:

${\displaystyle {\frac {dh_{W}}{dt}}=-D-rh_{W}}$

where the second contribution on the right-hand is due to the ocean adjustment damping relative to boundary layer and lateral processes.[12]

Further considerations can be made about the ocean heat budget. The heat budget is considered to be open only when a temperature anomaly appears on the eastern side of the ocean. In order for the temperature in the eastern Pacific to change, both a zonal velocity anomaly due to wind stress anomaly and vertical advection are needed.

In the first case, the horizontal velocity of heat transporting flow can be considered to be proportional to the wind stress anomaly as: ${\displaystyle U=\gamma \tau }$. This relation is valid as long as the basin taken into consideration is assumed to be purely wind-driven and not influenced by Earth rotation.[13] Positive ${\displaystyle U}$ values correspond to positive wind stress anomaly values.

Thereby, the temperature anomaly along the eastern Pacific (advection of the background climatological temperature field) can be seen as:

${\displaystyle -u{\frac {\partial T}{\partial x}}\sim U{\frac {\Delta _{h}T}{L}}}$

The figure shows the simulation of the oscillation of temperature anomaly and thermocline depth anomaly over a 40 years period of time. The amplitude of the oscillations decline with time due to damping effects included in the model. Starting conditions (T=2 °C, h_W=0).

where ${\displaystyle \Delta _{h}T}$ is the difference in temperature between the eastern and the western parts of the basin.

As far as convection is concerned, the vertical convection in the East can be normally estimated as

${\displaystyle -u{\frac {\partial {\bar {T}}}{\partial z}}\sim -{\bar {\omega }}{\frac {{\bar {T}}_{SURF}-{\bar {T}}_{-H}}{H}}}$

where the bar above the temperature refers to the climatological situation.[12] In this case, ${\displaystyle \omega }$ refers to the upwelling and its relation with the wind stress can be parameterised as ${\displaystyle {\bar {\omega }}=-\alpha {\bar {\tau }}}$ , where the minus sign ensures that to a positive wind stress corresponds a decrease in the upwelling on the eastern pool. In fact, a positive wind stress anomaly generates a corresponding negative upwelling anomaly ${\displaystyle {\tilde {\omega }}}$. The resulting reduction of deep cold water generates a temperature anomaly and thus a positive heat flux calculated as:

${\displaystyle -{\tilde {\omega }}{\frac {T_{SURF}-T_{-H}}{H}}\sim \alpha \tau {\frac {\Delta _{v}T}{H}}}$

where ${\displaystyle \Delta _{v}T}$ is the vertical temperature difference.

Since the surface water temperature is higher than the deep water temperature (${\displaystyle \Delta _{v}T>0}$), this contribution is positive for a positive wind stress anomaly.

Nonetheless, the downward shifting of the water temperature profile (the thermocline deepens by ${\displaystyle h_{E}}$) implies that the temperature at a depth ${\displaystyle -H}$ should be considered to be the climatological value found at ${\displaystyle -H+h_{E}}$. It is important to highlight that the surface temperature, at this point, includes the increase due to the anomaly.

Therefore, the following result is obtained:[12]

${\displaystyle -{\bar {\omega }}({\frac {T_{E}}{z}}-{\frac {\partial {\bar {T}}}{\partial z}}{\frac {h_{E}}{H}})\sim -{\frac {\bar {\omega }}{H}}T_{E}+{\frac {{\bar {\omega }}\Delta _{v}T}{H^{2}}}h_{E}}$.

The figure shows the simulation of relation between the temperature anomaly and the thermocline depth anomaly and how it changes over time. Same starting conditions as above.

Grouping the three contributions from advection, the variation over time of the temperature anomaly due to advection becomes:

${\displaystyle {\frac {dT_{E}}{dt}}=-(r'+{\frac {\bar {\omega }}{H}})+{\frac {{\bar {\omega }}\Delta _{v}T}{H^{2}}}h_{E}+(\gamma {\frac {\Delta _{h}T}{L}}+\alpha {\frac {\Delta _{v}T}{H}})\tau }$.

In the final contribution, a damping component (first one on the right-hand side) is added in a similar way to what has been done above for ${\displaystyle {\frac {dh_{W}}{dt}}}$.

It is possible to further assume that the relation between the wind stress anomaly and the temperature anomaly is given by: ${\displaystyle \tau =\mu T_{E}}$ .

Finally, the coupled model is complete and described as follows:

${\displaystyle {\frac {dh_{W}}{dt}}=-{\frac {\mu T_{E}}{\rho _{0}\beta _{0}R_{EQ}^{2}}}-rh_{W}}$

${\displaystyle {\frac {dT_{E}}{dt}}=[\mu (\gamma {\frac {\Delta _{h}T}{L}}+\alpha {\frac {\Delta _{v}T}{H}}+{\frac {{\bar {\omega }}L\Delta _{v}T}{H^{3}g'\rho _{0}}}-r')-{\frac {\bar {\omega }}{H}}]T_{E}+{\frac {{\bar {\omega }}\Delta _{v}T}{H^{2}}}h_{W}}$.

Despite the improvements, the previous model is still a simplification of the real mechanism that is much more complex in its behaviour. The animation clearly shows an elliptical behaviour over time in the relation between temperature and depth anomalies that is not observed in historical data observations.[14] The model described above considers a symmetrical behaviour for the two different phases (El Niño and La Niña), which is not what is observed in reality.[15] For instance, as shown in the work of McPhaden et al. (2000): "air–sea fluxes, which are a negative feedback on SST anomaly growth in the equatorial cold tongue are more effective at heating the ocean during cold phases of ENSO than they are at cooling the ocean during warm phases of ENSO. Alternately, the ability of upwelling and vertical mixing to cool the surface may saturate at some threshold beyond which further thermocline shoaling does not lead to further SST cooling".[15]