3.2 Form and growth mechanism of coupled mode

The complexity of (41) stems from the several different processes that play a role in the SST equation (20). To gain an understanding of the physics of the coupling, we must simplify the dispersion relation (41). We will now consider several different cases, including only one or two terms in the SST equation in turn to study their influence in isolation. We will begin with the simplest case which illustrates the coupled interaction, and then consider other processes which modify this underlying mechanism.

   
3.2.1 SST Case 1: Entrainment

The simplest case is the one where entrainment dominates the SST equation, and advection, air-sea flux, and tendency are small. Then (20) reduces to

$$0 = -\gamma_e(\mbox{SST}- \theta_{\mbox{\scriptsize {sub}}})$$

$$\mbox{SST}= \theta_{\mbox{\scriptsize {sub}}} = r_o \psi_o$$ (42)

implying perfect communication between thermocline perturbations and SST. Dominance of entrainment requires that $\gamma_e \gg ak$, $\gamma_e \gg \sigma$, $\gamma_e \gg m \gamma_o$ (numerical values are considered in section 4). Then the first term on the left side of (40) reduces to 1, and there is only one solution to the now linear equation for $\sigma$:

$$\sigma =\omega_r-r\left[ \alpha \kappa ^{2}L_o^2\left( \frac{... ...Gamma }+i}{ \left( \frac{\nu }{\Gamma }\right) ^{2}+1} \right]$$ (43)

The waves of our system move in a phase-locked fashion through the ocean and atmosphere. Because the dynamical ocean is the only prognostic field (the SST tendency term has been neglected), from one perspective the fluctuations exist fundamentally in the ocean. They are manifest in the atmosphere because it responds to the modification of $\mbox{SST}$ (and hence thermal forcing) induced by the ocean. But the ocean only moves because the atmosphere blows over it -- thus our mode is a coupled one.

We see the ocean connection by the presence of the oceanic Rossby wave frequency $\omega_r$ in (43). The second term in (43) contains a real part created by air-sea interaction which (slightly) slows down or speeds up the oceanic Rossby waves. But $\sigma$ also has an imaginary part:

$$Im(\sigma)=-r \alpha \kappa ^{2}L_o^2(\frac{\mu }{2}+1) \frac{1}{\left( \frac{\nu }{\Gamma }\right) ^{2}+1}$$ (44)

Since the waves have the form $e^{i(kx-\sigma t)}\sin{ly}$, then $Im(\sigma )$ must be positive for growth. All the variables in (44) are positive-definite except $(\frac{\mu }{2}+1)$. For $Im(\sigma)>0$, we need $\mu /2+1<0$. What is the physical meaning of this condition on $\mu$? It arose from the ``surface windstress'' term in the oceanic forcing (17). Since

$$\psi _{s}=\frac{1}{2}\widehat{\psi}-\widetilde{\psi}=-(\frac{\mu }{2}+1) \widetilde{\psi}$$

surface streamfunction anomalies have the same sign as the vertical shear $\widetilde{\psi}$ when $\mu < -2$: i.e. the waves are then `equivalent barotropic'.

Waves near barotropic resonance ( $\widehat{U}\approx \frac{\widehat{\beta }}{\kappa ^{2} }$, with $\vert\mu \vert$ large) exhibit the strongest barotropic response, and therefore grow the fastest. But the growth rates also depend on the size of the equilibration term $\Gamma$ relative to the advection-propagation parameter $\nu$; $\nu$ depends on $\widehat{U}$, $\widetilde{U}$, and the wave size. When $\vert\nu\vert \ll \Gamma$, the wave has time to equilibrate with the oceanic forcing (i.e., the left- and right-hand sides of (25) independently approach zero). A large response will be excited, enhancing the coupling. But if advection-propagation is much more rapid than equilibration ( $\vert\nu\vert \gg \Gamma$), the response of the atmosphere is smaller and shifted away from the oceanic SST anomaly, and growth of the coupled mode is slowed. These effects are encapsulated in the factor $\left(\left( \frac{\nu }{\Gamma }\right) ^{2}+1\right)^{-1}$ in (44). It is the equilibrated atmospheric modes that couple most efficiently and grow most rapidly.

The structure of the fastest-growing mode for the entrainment-dominated SST case is sketched in Figure 4. As described above, any mode with positive growth rate must have $\mu < -2$, so the atmospheric response is equivalent barotropic ( $\vert\psi_1\vert >\vert\psi_2\vert>\vert\psi_s\vert$ and each has the same sign), weakest at the surface and strongest aloft. If the surface pressure anomaly is positive, the resultant anticyclonic surface winds will cause downward Ekman pumping in the ocean which deepens the already-deep thermocline leading (see (42)) to a warmer surface and a positive feedback. If the surface pressure anomaly is negative, Ekman dynamics will suck up the thermocline resulting in anomalously cold winter SST, again a positive feedback. For the coupling physics adopted here, coupled growth will occur whenever the atmospheric response is equivalent barotropic.

  

Figure 4: Phase relationships between ocean and atmosphere for the fastest growing coupled mode. The symbols H and L denote highs and lows of atmospheric pressure, with the amplitude of the pressure anomaly increasing with height. The atmospheric response is `equilibrated', as in figure 2b. The symbols W and C denote warm and cold SST, and the undulating line indicates the depth of the thermocline. Note the high (low) pressure above warm (cold) water, and the phase match between wind stress and current.

The atmospheric and oceanic wave components need not be in phase with one another, and the degree of phase-matching determines the rate at which the coupled mode grows. Growth is fastest when $\nu /\Gamma$ is small in (44), which (from (34) and (42)) occurs when the atmosphere equilibrates completely with the underlying ocean, and high pressures occur directly over warm, deep-thermocline water ( $\widetilde{\psi} \propto \mbox{SST}\propto \psi_o$). Then the Ekman pumping acts directly to increase the amplitude of thermocline perturbations; the wind applies torque to the ocean to reinforce the existing circulation. As the advection/propagation term $\vert\nu \vert$ increases, the atmospheric perturbation is ``blown away'' from the oceanic anomaly which generates it, resulting in a phase lead or lag; the Ekman pumping no longer perfectly matches the location of greatest anomaly, so growth is slower. When $\vert\nu \vert$ completely dominates $\Gamma$, the phase shift is 90$^{\circ}~$( $\widetilde{\psi}\propto i\mbox{SST}$; $\widetilde{\psi}\propto i\psi_o$). In this case, the Ekman pumping does not increase the thermocline anomalies at all because the wind forcing is in quadrature with the ocean response. These two cases (zero lag and quadrature) correspond to the equilibrated and directly-forced modes shown in figure 2. More specifically, the atmospheric wave lies westward of the oceanic wave by a phase angle:

$$\theta =\mbox{Tan}^{-1}\left( \frac{\nu }{\Gamma }\right)$$ (45)

If $\nu >0$, atmospheric pressure crests lie eastward of SST maxima, and vice versa for $\nu <0$.

For atmosphere-ocean phase shifts between 90$^{\circ}~$and -90$^{\circ}~$, in the growing mode the circulation induced by oceanic thermal forcing yields a windstress which reinforces the sense of the pre-existing circulation. If the waves are able to equilibrate with their energy source $(\vert\nu \vert\ll \vert\Gamma \vert)$, growth is rapid and the atmospheric geopotential anomalies lie directly over their SST sources. But if the waves in the atmosphere propagate away from the energy source more rapidly than that source can be renewed $(\vert\nu \vert\gg \vert\Gamma \vert)$, the coupled phenomenon grows slowly, with atmospheric waves shifted downstream from their SST sources (see (34)). In all cases of growth, though, the atmospheric anomaly hovers near the SST heat source.

It is useful to draw an analogy with a burning candle. The heat of the flame melts and vaporizes the wax directly below it, which then provides chemical energy to allow the flame to grow and maintain itself. If we blow gently on the candle flame, we may transport it away from its fuel source faster than the fuel is renewed: the flame weakens, and may die if we blow hard enough. In all cases, though, the flame hovers above or beside the wick.

   
3.2.2 SST Case 2: Entraintment & Tendency

What happens if we include the SST tendency term in equation (20), but still neglect meridional advection (and therefore a k in (41))? In the limit where $a k \ll \gamma_e$, (41) reduces to

$$\sigma = \frac{1}{2}\left(\omega_r - i \gamma_e\right) \pm i\... ...\Gamma }+i}{\left( \frac{\nu }{\Gamma }\right) ^{2}+1}\right]}$$ (46)

In the case where entrainment is much faster than Rossby propagation ( $\gamma_e \gg \omega_r$) and is also faster than the air-sea coupling ( $\gamma_e \gg r[~]$), we may use the approximation $\sqrt{1+x} \approx 1+x/2$ to find the approximate solutions:

$$\sigma_1 \approx \omega_r - r\alpha\kappa^{2}L_o^2\left(\frac... ...ac{\nu }{\Gamma }+i}{\left( \frac{\nu }{\Gamma }\right)^{2}+1}$$ (47)

$$\sigma_2 \approx - i \gamma_e + r\alpha\kappa^{2}L_o^2 \left(... ...ac{\nu}{\Gamma }+i}{\left( \frac{\nu }{\Gamma }\right) ^{2}+1}$$ (48)

The first solution is identical to the entrainment solution without the tendency term (43), described in detail in section 3.2.1. The second solution is dominated by rapid SST damping through entrainment (i.e., by the $-i\gamma_e$ term). The Rossby wave propagation term canceled in the expression for $\sigma_2$: the solution does not propagate as a Rossby wave, and is, in fact, decoupled from the dynamic ocean: therefore we call it an ``SST-only'' mode. The second term, describing the air-sea interaction, has the opposite sign in the SST-only mode as in the ``entrainment mode'' discussed in section 3.2.1, suggesting that the conditions for growth discussed there cause enhanced decay in this mode.

The structure of the SST-only mode is quite simple, and is depicted in figure 5. We begin with a warm patch of SST, but with only a slightly perturbed thermocline having the opposite sign as SST. The SST patch generates an atmospheric response above or downstream from it (depending on $\nu /\Gamma$), but the patch is rapidly damped by the $\gamma_e(\mbox{SST} - \theta_{\mbox{\scriptsize {sub}}})$ term in (20), and decays in a short time $1/\gamma_e$. The slight Ekman pumping supplied by the wind during that time acts only to diminish the initial thermocline anomaly; thus all fields decay to zero rapidly.

  

Figure 5: Configuration of the rapidly-damped SST-only mode (equation (48)). SST is out of phase with the very small subsurface thermal anomalies, leading to rapid damping of SST.

The two solutions span the range of possible initial conditions for SST and $\psi _{o}$. If we begin with an arbitrary pattern of SST and $\psi _{o}$, the component which has SST and $\psi _{o}$ in phase will grow and propagate as described in section 3.2.1 (assuming conditions for growth are met), while the out-of-phase component will decay rapidly via the process described here, until only the in-phase component is observed.

   
3.2.3 SST Case 3: Advection & Tendency

Even though our SST scaling analysis suggests that entrainment is at least as important as advection in winter months, it is useful to consider the advection mechanism in isolation. Accordingly, we consider the form of the SST equation (20) with $\gamma_e \rightarrow 0$ and $\gamma_o \rightarrow 0$.

$$\frac{\partial}{\partial t}\mbox{SST}' = - \vec{u'}\cdot\nabla\overline{\mbox{SST}}$$

In the same limit, the dispersion relation (41) becomes

$$\sigma = \frac{1}{2}\omega_r \pm i\sqrt{-\frac{1}{4}\omega_r^... ...\Gamma }+i}{\left( \frac{\nu }{\Gamma }\right) ^{2}+1}\right]}$$ (49)

As before, we consider the case where the coupling term rak[ ] is smaller than the Rossby wave propagation term $\omega_r$, in which case we get the following two approximate solutions:

$$\sigma_1=\omega_r - \frac{r a k}{\omega_r} \left[\alpha\kapp... ...}{\Gamma }+i}{\left( \frac{\nu }{\Gamma }\right)^{2}+1}\right]$$ (50)

$$\sigma_2 = \frac{r a k}{\omega_r} \left[\alpha\kappa^{2}L_o... ...}{\Gamma }+i}{\left( \frac{\nu }{\Gamma }\right)^{2}+1}\right]$$ (51)

The solution $\sigma_1$ has exactly the same structure as the entrainment mode described in section 3.2.1, with r replaced by $r a k/\omega_r$. Growth occurs in this ``advection mode'' when the atmosphere responds with barotropic highs over warm water, exactly as in section 3.2.1.

Like the entrainment mode, the advection mode has warm SST where $\psi _{o}$ is large (see figure 4), but for an entirely different reason, illustrated in figure 6. Oceanic streamfunction anomalies will propagate from east to west. A streamfunction high (depressed thermocline) will generate a northward flow to its west, advecting warm water from the south and creating a warming trend there. When the $\psi _{o}$ anomaly propagates to that spot, the advection ceases, and so does the warming. When the $\psi _{o}$ anomaly continues on to the west, it generates southward flow, bringing cold water which cools the SST patch. Therefore, a maximum in SST is observed at the maximum in $\psi _{o}$, and appears to follow that maximum as it propagates westward. SST and $\psi _{o}$ are in phase, and waves which propagate more slowly have more time to build up larger SST anomalies: this is why the Rossby-wave propagation term occurs in the denominator of the second term in $\sigma_1$.

  

Figure 6: The process by which advection of mean meridional SST gradient leads to warm SST anomalies over deep-thermocline water. See text for full description.

The second solution has no Rossby-wave propagation, and SST and $\psi _{o}$ are out of phase. The solution is most strongly damped when air-sea coupling is strong.

   
3.2.4 SST Case 4: Air-sea flux, entrainment, and tendency

The inclusion of the surface flux term into the SST equation should reduce the growth of the coupled mode: after all, if a warm patch of SST is losing heat to the atmosphere at a rate comparable to the rate of heating by entrainment or advection, the anomaly will have smaller magnitude and thus generate a less powerful atmospheric circulation. However, the most rapidly growing mode from the previous three cases is unaffected by the air-sea flux term. Our fastest-growing mode has $\nu /\Gamma$ = 0, so m=0 in (35): $\mbox{SST}= \theta_a = r_a \widetilde{\psi}$: there is no air-sea temperature difference (complete equilibration), so the surface heat flux shuts off. In fact, by setting m=0 in (40), we get (46) when advection is small.

We now consider the case where m is nonzero, but for convenience we assume advection is small ( $a k \ll \gamma_e$); our results will also hold for non-negligble a k. In the limit $m\gamma_o \gg \omega_r$ and $m\gamma_o \gg r[~]$, (41) can be approximated by:

$$\sigma_1 \approx \omega_r - \frac{\gamma_e}{\gamma_e + m\gamm... ...ac{\nu }{\Gamma }+i}{\left( \frac{\nu }{\Gamma }\right)^{2}+1}$$ (52)

$$\sigma_2 \approx - i \gamma_e -i m\gamma_o + \frac{\gamma_e}{... ...ac{\nu}{\Gamma }+i}{\left( \frac{\nu }{\Gamma }\right) ^{2}+1}$$ (53)

These two modes closely resemble the entrainment modes discussed in section 3.2.2; however, the coupled growth term of the coupled solution ($\sigma_1$) is multiplied by the factor $\gamma_e/(\gamma_e + m\gamma_o)$, and damping of the ``SST-only'' solution ($\sigma_2$) is enhanced by the air-sea flux. If $\gamma_e \approx \gamma_o$ (typical of the annual average), growth off-resonance (where $\vert\vert m\vert\vert\sim 1$) is reduced by about a factor of two. During the winter, when $\gamma _e$ is larger than $\gamma_o$, growth will not be significantly affected. During the summer, when $\gamma_e \sim 0$, (52) and (53) reduce to

$$\sigma_1 \approx \omega_r \qquad \sigma_2 \approx -i m \gamma... ... \frac{\nu/\Gamma - i(\nu/\Gamma)^2}{1+(\nu/\Gamma)^2}\gamma_o$$ (54)

Coupling between the geostrophic ocean and the mixed layer has ceased entirely; the first solution takes the form of uncoupled propagating oceanic Rossby waves with no expression in the mixed layer or atmosphere; the second equation shows the effect of a 2-layer QG atmosphere over a ``swamp'' mixed layer. This mode resembles the ``QG atmosphere over a copper plate'' discussed by Frankignoul (1985): it is characterized by rapidly-damped patterns in SST and atmosphere which propagate eastward or westward depending on the phase of the atmosphere's response to SST. If warm SST produces warm air to the east of the SST anomaly ( $\nu/\Gamma >0$), this warmth results in a heat flux back into the ocean farther east than it originated, resulting in eastward phase propagation, and vice versa for westward phase shifts. However, since this ``heat flux'' mode is always damped on a timescale of order $\gamma_o^{-1} \sim$ 8 months, it is unlikely to play a role in decadal variability.

Allowing air-sea flux to affect the mixed layer cannot destroy our growing mode, because the fastest-growing mode has vanishingly small air-sea flux. However, it may reduce growth rates somewhat when conditions are slightly off-resonance. When air-sea flux dominates over entrainment (as might happen in summer), the mixed layer decouples from the dynamic ocean; Rossby waves continue to propagate in the thermocline while the mixed layer exhibits rapidly-damped air-sea interaction as described by Frankignoul (1985).