Seasonal variation of the M ₂ tide

Abstract

The seasonal cycle of the main lunar tidal constituent M ₂ is studied globally by an analysis of a high-resolution ocean circulation and tide model (STORMTIDE) simulation, of 19 years of satellite altimeter data, and of multiyear tide-gauge records. The barotropic seasonal tidal variability is dominant in coastal and polar regions with relative changes of the tidal amplitude of 5–10 %. A comparison with the observations shows that the ocean circulation and tide model captures the seasonal pattern of the M ₂ tide reasonably well. There are two main processes leading to the seasonal variability in the barotropic tide: First, seasonal changes in stratification on the continental shelf affect the vertical profile of eddy viscosity and, in turn, the vertical current profile. Second, the frictional effect between sea-ice and the surface ocean layer leads to seasonally varying tidal transport. We estimate from the model simulation that the M ₂ tidal energy dissipation at the sea surface varies seasonally in the Arctic (ocean regions north of 60°N) between 2 and 34 GW, whereas in the Southern Ocean, it varies between 0.5 and 2 GW. The M ₂ internal tide is mainly affected by stratification, and the induced modified phase speed of the internal waves leads to amplitude differences in the surface tide signal of 0.005–0.0150 m. The seasonal signals of the M ₂ surface tide are large compared to the accuracy demands of satellite altimetry and gravity observations and emphasize the importance to consider seasonal tidal variability in the correction processes of satellite data.

Appendices

Appendix 1

Assume that the heat (freshwater) flux equations in the upper modeled ocean layer are given by

$$ \frac{dP}{dt}=A, $$

(11)

where P represents the modeled quantity, i.e., temperature (salinity). The function A represents the heat (freshwater) flux explicitly added by surface restoring and is usually described by

$$ A(t)=\kappa(P_{0}(t)-P(t)) $$

(12)

$$ \kappa=\frac{1}{\tau}. $$

(13)

where P ₀(t) represents the climatological data, τ is the timescale, and κ is the coupling coefficient. Combining Eqs. 11 and 12 yields

$$ \frac{dP}{dt}=\kappa(P_{0}(t)-P(t)). $$

(14)

We use monthly climatology, and thus, it is advisable to use the “time lag and amplitude compensation for Newtonian fluxes” (Cherniawsky and Holloway 1991).

Assume that P ₀ is a seasonal modulated quantity with the annual frequency ω _s ,

$$ P_{0}(t)=\bar{P}_{0} + \hat{P}_{0}\cdot e^{i\omega_{s} t}. $$

(15)

The surface flux is written as follows:

$$ A(t)= \kappa(c P_{0}(t+\Delta t)-P(t)), $$

(16)

where c is the amplitude, and Δt is the phase correction, and write them as follows:

$$ c^2=1+\left(\frac{\omega_{s} }{\tau}\right)^{2} $$

(17)

$$ \Delta t=\frac{1}{\omega_{s}}\tan^{-1}\left(\frac{\omega_{s}}{\tau}\right). $$

(18)

For our model simulation, we assume a timescale of τ=90 days, which yields c=1.8 and Δt = 58 days.

Appendix 2

We assume our tidal signal ζ(t) is a superposition of three oscillations with frequencies ω _M2, ω _a , and ω _b and phases ϕ _M2, ϕ _b , and ϕ _b . The indices a and b refer to the annual satellites α ₂ and β ₂. Further, the frequencies are related through

$$ \omega_{a}=\omega_{M2}-\omega_{s} $$

(19)

$$ \omega_{b}=\omega_{M2}+\omega_{s}, $$

(20)

where ω _s is the frequency of the seasonal cycle, and thus,the two constituents α ₂ and β ₂ are representing annual satellites of the M ₂ tide. We can superpose the tidal signal at a time t and write

$$\begin{array}{@{}rcl@{}} \zeta(t)&=&A_{M2} \Re(\exp[i(\omega_{M2}(t-t_0)+V_{M2}(t_0)-\phi_{M2})]\\ &&+A_{a} \Re(\exp[i(\omega_{a}(t-t_0)+V_{a}(t_0)-\phi_{a})]\\ &&+A_{b} \Re(\exp[i(\omega_{b}(t-t_0)+V_{b}(t_0)-\phi_{b})] \end{array} $$

(21)

where V _k is the astronomical arguments for constituent k and reference time t ₀, and A _k and ϕ _k are the amplitude and phase lag of a constituent k, respectively. We further define the argument of M ₂ as arg _M2 = ω _M2(t − t ₀) + V _M2(t ₀) − ϕ _M2 and rewrite the tidal signal:

$$\begin{array}{@{}rcl@{}} \zeta(t) &=& (A_{M2} + S(t) ) \cos(arg_{M2})\\ S(t) &=& A_{a} \cos(-\omega_{s} (t-t_0) +(V_{a}(t_0)-V_{M2}(t_0))\\&&-(\phi_{a}-\phi_{M2})) \\ &+& A_{b} \cos(\omega_{s} (t-t_0) +(V_{b}(t_0)-V_{M2}(t_0))\\ &&-(\phi_{b}-\phi_{M2})) \end{array} $$

(22)

This can be interpreted as an amplitude modulation S(t) of the carrier wave A _M2 cos(arg _M2).

Appendix 3

Table 2 lists the tidal constituents used in the harmonic analysis of along-track altimeter data. Following Cherniawsky et al. (2010) and to minimize the effects of aliasing and of strong nontidal signals, we adopted the following edit criteria when tabulating or plotting the computed amplitudes A and Greenwich phase lags G of the analyzed constituents.

• 1. We require the analyzed amplitudes to be larger than their error estimates: A > σ.

• 2. We require the computed complex amplitudes C = A exp(i G) on ascending and descending passes to agree near pass crossovers. Using r = 2 | C _{a s c} − C _{d e s c} | / | C _{a s c} + C _{d e s c} | , where C _{a s c} and C _{d e s c} are average complex amplitudes at two ascending and two descending along-track locations near a crossover, we exclude constituents near a crossover and up to half the distance between the crossovers, when r > 0.5.

• 3. We exclude locations where the root mean square sea level variability of the residual signal after detiding exceeds 0.12 m.

Table 2 Tidal constituents selected for harmonic analysis, listed in order of frequency, with Doodson numbers and main (lowest-frequency) alias periods T _al

These edit criteria remove most, though not all, suspect data. The number of remaining valid data locations (N _v in Table 2) depends on the relative amplitudes and susceptibility to aliasing of each constituent (Ray 1998; Cherniawsky et al. 2001, 2010). However, we only used the first criterion for the four strongest constituents O ₁, K ₁, M ₂, and S ₂.