Synoptic eddy feedback and air–sea interaction in the North Atlantic region

Abstract

This paper explores the role of synoptic eddy feedback in the air-sea interaction in the North Atlantic region, particularly the interaction between the North Atlantic Oscillation (NAO) and the North Atlantic sea surface temperature anomalies (SSTA) tripole. A linearized five-layer primitive equation atmospheric model with synoptic eddy and low-frequency flow (SELF) interaction is coupled with a linearized oceanic mixed-layer model to investigate this issue. In this model, the “climatological” storm track/activity (or synoptic eddy activity) is characterized in terms of spatial structures, variances, decay time scales and propagation speeds through the complex empirical orthogonal function (CEOF) analysis on the observed data, which provides a unique tool to investigate the role of synoptic eddy feedback in the North Atlantic air–sea coupling. Model experiments show that the NAO-like atmospheric circulation anomalies can produce tripole-like SSTA in the North Atlantic Ocean, and the tripole-like SSTA can excite a NAO-like dipole with an equivalent barotropic structure in the atmospheric circulation, which suggests a positive feedback between the NAO and the SSTA tripole. This positive feedback makes the NAO/SSTA tripole-like mode be the leading mode of the coupled dynamical system. The synoptic eddy feedback plays an essential role in the origin of the NAO/SSTA tripole-like leading mode and the equivalent barotropic structure in the atmosphere. Without synoptic eddy feedback, the atmosphere has a baroclinic structure in the response field to the tripole-like SSTA forcing, and the leading mode of the dynamic system does not resemble NAO/SSTA tripole pattern.

Appendix-The quasi-geostropic model

The quasi-geostrophic (QG) model mentioned in the Sect. 3 is a linearized two-layer model in middle latitude. It starts from the QG potential vorticity equation,

$${\frac{{\partial P}}{{\partial t}} + J(\psi, P) = F,}$$

(A1)

where P denotes potential vorticity, ψ means streamfunction, J represents Jacobi term and F refers to the forcing term (e.g., linear damping). If we separate potential vorticity P into three parts: climatological basic state ${\bar{P}_{c},}$ low-frequency anomaly ${\bar{P}_{a},}$ and high frequency synoptic eddy P′, then the low-frequency anomaly of ${\bar{P}_{a}}$ can be written as

$${\frac{{\partial \bar{P}_{a}}}{{\partial t}} + J(\bar{\psi}_{a},\bar{P}_{c}) + J(\bar{\psi}_{c}, \bar{P}_{a}) +\overline{{J({\psi}^{\prime},{P}^{\prime})}} |_{a} = \bar{F}_{a},}$$

(A2)

where ${\overline{{J({\psi}^{\prime},{P}^{\prime})}} |_{a}}$ is related to synoptic eddy forcing. Assume the climatological streamfunction

$${\bar{\psi}_{c} = - Cy,}$$

(A3)

where C denotes a constant, and y represents the axis of meridional direction. The equation related to zonal-mean zonal wind anomaly can be written as

$${\frac{{\partial [\bar{P}_{a} ]}}{{\partial t}} +[\overline{{J({\psi}^{\prime},{P}^{\prime})}} |_{a} ] = [\bar{F}_{a}],}$$

(A4)

where “[]” represents the zonal mean. For simplicity, the “[]” is dropped in the following equations, but the equations are still written in terms of zonal-mean sense.

The two-layer QG model can be written as

$${\frac{{\partial \bar{P}_{{a1}}}}{{\partial t}} +\overline{{J({\psi}^{\prime},{P}^{\prime})}} |_{{a1}} = \bar{F}_{{a1}},}$$

(A5)

$${\frac{{\partial \bar{P}_{{a2}}}}{{\partial t}} +\overline{{J({\psi}^{\prime},{P}^{\prime})}} |_{{a2}} = \bar{F}_{{a2}},}$$

(A6)

where the lower-level and upper-level potential vorticity in pressure coordinate can be expressed as

$${\bar{P}_{{a1}} = \nabla^{2} \bar{\psi}_{{a1}} - f^{2}_{0}\frac{{\bar{\psi}_{{a1}} - \bar{\psi}_{{a2}}}}{{\sigma (\Delta p)^{2}}},}$$

(A7)

$${\bar{P}_{{a2}} = \nabla^{2} \bar{\psi}_{{a2}} + f^{2}_{0}\frac{{\bar{\psi}_{{a1}} - \bar{\psi}_{{a2}}}}{{\sigma (\Delta p)^{2}}},}$$

(A8)

where f ₀ denotes Coriolis parameter, σ is the static stability parameter, and Δp represents the pressure difference of the two layers.

The forcing term includes the linear damping related to potential vorticity and external heat forcing F _T related to temperature, therefore,

$${\bar{F}_{{a1}} = - d_{1} \nabla^{2} \bar{\psi}_{{a1}} - d_{T}(\bar{\psi}_{{a1}} - \bar{\psi}_{{a2}} - F_{T}),}$$

(A9)

$${\bar{F}_{{a2}} = - d_{2} \nabla^{2} \bar{\psi}_{{a2}} - d_{T}(\bar{\psi}_{{a1}} - \bar{\psi}_{{a2}} - F_{T}),}$$

(A10)

where d ₁ and d ₂ are linear friction coefficients for lower and upper levels. d _T denotes thermal damping coefficient.

Combining Eqs. (A5), (A6), (A9) and (A10), we have,

$${\frac{{\partial \bar{P}_{{a1}}}}{{\partial t}} +\overline{{J({\psi}^{\prime},{P}^{\prime})}} |_{{a1}} = - d_{1}\nabla^{2} \bar{\psi}_{{a1}} - d_{T} (\bar{\psi}_{{a1}} -\bar{\psi}_{{a2}} - F_{T}),}$$

(A11)

$${\frac{{\partial \bar{P}_{{a2}}}}{{\partial t}} + \overline{{J(\psi,{P}^{\prime})}} |_{{a2}} = - d_{2} \nabla^{2} \bar{\psi}_{{a2}} + d_{T}(\bar{\psi}_{{a1}} - \bar{\psi}_{{a2}} - F_{T}),}$$

(A12)

If steady state is assumed and adding (A11) and (A12) together, the following relationship can be obtained,

$${\overline{{J({\psi}^{\prime},{P}^{\prime})}} |_{{a1}} +\overline{{J({\psi}^{\prime},{P}^{\prime})}} |_{{a2}} = - d_{1}\nabla^{2} \bar{\psi}_{{a1}} - d_{2} \nabla^{2} \bar{\psi}_{{a2}},}$$

(A13)

Letting

$${F_{e} = \nabla^{{- 2}}\{\overline{{J({\psi}^{\prime},{P}^{\prime})}}|_{{a1}} +\overline{{J({\psi}^{\prime},{P}^{\prime})|_{{a2}}}}\}}$$

(A14)

Eq. (A13) changes to

$${\nabla^{2} (- d_{1} \bar{\psi}_{{a1}} - d_{2} \bar{\psi}_{{a2}} - F_{e})= 0,}$$

(A15)

therefore,

$${- d_{1} \bar{\psi}_{{a1}} - d_{2} \bar{\psi}_{{a2}} - F_{e} = B_{1} x +B_{2} y + B_{0},}$$

(A16)

where B ₀, B ₁ and B ₂ are any constants and x and yrepresent zonal and meridional axes. For the zonal symmetric case, B ₁ equals to zero. Taking the derivative in meridional direction to (A16),

$${u_{{a1}} = - d_{2} u_{{a2}} /d_{1} + F_{{ey}} /d_{1} + B_{2} /d_{1},}$$

(A17)

where u _a1 and u _a2 are the zonal-mean zonal wind at lower and upper layers, ${F_{{ey}} = \frac{{\partial F_{e}}}{{\partial y}}.}$ Since u _a1 is zero if the lower level is at earth surface, B ₂ can be zero or d ₂ u _a2 − F _ey , and d ₂ u _a2 − F _ey is not a valid solution; therefore, B ₂ must be zero. The relationship between the zonal-mean zonal wind at lower and upper layers can be derived,

$${u_{{a1}} = - d_{2} u_{{a2}} /d_{1} + F_{{ey}} /d_{1},}$$

(A18)

which is used as Eq. (14) in Sect. 3. The u _a1 and u _a2 have opposite signs when synoptic forcing equals zero and it has a baroclinic structure in the vertical. If the synoptic eddy forcing has a positive feedback on mean flow and it is strong enough, u _a1 and u _a2 can have the same sign and have an equivalent barotropic structure in the vertical. This argument is also applicable to stationary wave and zonal flow interaction.