Response of sea surface temperature to atmospheric rivers

Atmospheric rivers (ARs), responsible for extreme weather conditions, are mid-latitude systems that can cause significant damage to coastal areas. While forecasting ARs beyond two weeks remains a challenge, past research suggests potential benefits may come from properly accounting for the changes in sea surface temperature (SST) through air–sea interactions. In this paper, we investigate the impact of ARs on SST over the North Pacific by analyzing 25 years of ocean reanalysis data using an SST budget equation. We show that in the region of strong ocean modification, ocean dynamics can offset over 100% of the anomalous SST warming that would otherwise arise from atmospheric forcing. Among all ocean processes, ageostrophic advection and vertical mixing (diffusion and entrainment) are the most important factors in modifying the SST tendency response. The SST tendency response to ARs varies spatially. For example, in coastal California, the driver of enhanced SST warming is the reduction in ageostrophic advection due to anomalous southerly winds. Moreover, there is a large region where the SST shows a warming response to ARs due to the overall reduction in the total clouds and subsequent increase in total incoming shortwave radiation.


Supplementary Note 1: Derivation of Mixed Layer Potential Temperature tendency
According to Forget, G. et al. (2015) 1 , the ECCO's on-transformed governing equation for potential temperature is where vectors are denoted in bold, ∇ z is the horizontal divergence operator with fixed z, (v, w) = v Eu , w Eu + v b , w b is the residual velocity with v Eu , w Eu being the Eulerian velocity and v b , w b the bolus velocity representing eddy mixing, F hdif and F vdif are the diffusive fluxes in horizontal and vertical directions, and F Θ = F sw + F lw + F sen + F lat is the sum of upward fluxes including shortwave radiation, longwave radiation, sensible heat flux and latent heat flux.Let H be the ocean depth and η = η (x, y,t) the sea surface height, z * -coordinate is defined as z * = (z − η) /s * where s * = 1 + η/H the geometrical factor.The transformed equation in the z * -coordinate is 1 The η evolves according to where PmE is the precipitation and evaporation freshwater flux, and w Eu η is the Eulerian vertical velocity at the sea surface.We then define the mixed-layer average operator The mixed-layer averaging comes with the following identities using fundamental theorem of calculus, Applying the mixed-layer averaging to (2), we derive where we substitute the tendency from (1) and use (3) to replace ∂ η/∂t.Using (5b) and letting (•) ′ = (•) − (•), we can rewrite the advection term in z-coordinates as where we also use the boundary condition w b η = 0 such that w η = w Eu η .We substitute (7) into (6) to obtain

Deriving the Bottom Boundary Flux
In this section, we introduce the method for obtaining the bottom boundary flux term in (8), which emerged in the identity (5a).For convenience, the variable (•) n denotes the information at time In the MITgcm, the potential temperature is stepped forward in time as 2 where ∆t is the interval used to do time-stepping, and G n+1/2 is the mean potential temperature tendency during the interval t ∈ [t n ,t n+1 ].We now write down the tendency of Θ as We substitute Θ n+1 with (10), we can rearrange (11) into The terms in the parenthesis can be rearranged as where the physics reads that This term is then counted as vertical mixing or detrainment according to the sign of (h − η) t .

Deriving Dilution Effect Θdilu
Deriving the dilution effect Θdilu needs two time-average products PmE Θ η − Θ = PmE Θ η − PmE Θ .The first term can be obtained explicitly with whose detail derivation can be found in the Section Derive PmE Θ η in MITgcm.For PmE Θ , we first expand each term with the decomposition The second term on the right-hand-side is a time-correlation term.Since it is not obvious to us that higher (lower) Θ implies higher (lower) PmE, we assume this term is negligible, meaning Therefore, we can compute Θdilu = 1

Derive PmE Θ η in MITgcm
The relations of heat budget variables in MITgcm 3 are listed below: where list more equations than we need for future reference.For open-ocean, i.e., when there is no sea ice: We use (19a), (19c), (19f), (19h), (20a), and (20b) to derive

2 .
Compute Θloc , Θsw , Θlw , Θsen , Θlat , Θhdif , Θvmix and (h − η) t .The computation of Θloc and (h − η) t is exact because ECCO releases the snapshot output.The values of Θhdif and Θhdif can also be computed exactly because ECCO releases the fluxes on grid faces.The fluxes at the bottom of the mixed layer is obtained by linearly interpolating the fluxes on the grid faces that encloses the desired depth.3. Compute Θdilu as explained in Section Deriving Dilution Effect Θdilu .4. Compute − Θ − Θ η−h ∂ (h − η) /∂t/h as explained in Section Deriving the Bottom Boundary Flux.