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Wave activity flux (Takaya–Nakamura)

Also called: Takaya-Nakamura flux, Takaya-Nakamura wave activity flux, TN01, Plumb flux, Rossby wave packet, Rossby wave propagation

The wave activity flux of Takaya and Nakamura (2001) is a vector that points along the group velocity of quasi-stationary Rossby waves: where wave energy is travelling, rather than where the ridges and troughs are. Its divergence marks a wave source, such as tropical convection or flow over a mountain range, and its convergence marks where the waves are depositing activity and the downstream flow will amplify days later. That makes it a practical tool for downstream development, blocking onset and the teleconnections that radiate from the tropics, such as the El Niño–PNA pathway.

The equation

For a basic state with horizontal wind $\mathbf{U}=(U,V)$ and a streamfunction anomaly $\psi'$, the horizontal flux on a pressure surface is

$$\mathbf{W}=\frac{p\cos\phi}{2\,|\mathbf{U}|}\begin{pmatrix}U(\psi_x'^2-\psi'\psi_{xx}')+V(\psi_x'\psi_y'-\psi'\psi_{xy}')\\[2pt]U(\psi_x'\psi_y'-\psi'\psi_{xy}')+V(\psi_y'^2-\psi'\psi_{yy}')\end{pmatrix}$$

It generalizes Plumb's (1985) flux for stationary waves on a zonal flow to a zonally varying basic state and to slowly migrating waves. It is phase-independent: it does not oscillate between ridge and trough, so packets can be followed without averaging over a wavelength.

How to read it

• Arrows arcing along the jet are a wave train; follow them to see where a disturbance will be felt next.
• Where the arrows converge and slow down, the wave amplifies: that is where ridges build and blocks form.
• A packet leaving the tropical Pacific toward North America is the tropical heating reaching the extratropics.

The live forecast

The flux is computed at 250 hPa, the level of the jet core and the stationary-wave maximum, for each of 25 AIFS ensemble members at every day to day 15, and then averaged. Because the flux is quadratic in $\psi'$, the flux of the ensemble mean would fade with lead time as the members decorrelate, while the mean of the members' fluxes keeps the packets that are actually forecast. Each member's $\psi'$ is smoothed with a 5-day running mean along the forecast, because the theory is for quasi-stationary waves. The basic state is the ERA5 1991–2020 climatology plus the 30-day mean anomaly of the AIFS analyses, and shading is hatched where fewer than 60 % of members agree on the sign of the divergence.