Interaction of the Gulf Stream with small scale topography: a focus on lee waves

The generation of lee waves in the Gulf Stream along the U.S. seaboard is investigated using high resolution realistic simulations. The model reproduces the surface signature of the waves, which compares favourably with observations from satellite sun glitter images in the region. In particular, a large number of internal waves are observed above the Charleston Bump. These waves match well with the linear theory describing topographically-generated internal waves, which can be used to estimate the associated vertical transport of momentum and energy extracted from the mean flow. Finally, small scale topographic features are shown to have a significant impact on the mean flow in this region of the Gulf Stream, and the specific role of lee waves in this context is outlined.


Introduction
The supplementary information below provide further details about methods used to describe the properties of the lee waves in the simulations.

Tools and methods used in the diagnostics of lee waves
This sections provides further details about the diagnostics performed on the simulation output.

(k, ω) spectrum calculation
Here we present how to compute the (k, ω) spectrum (dispersion diagram) of vertical velocity variance at 100 m depth (w 100 (t, x, y)).
Firstly, the spatial 2D Fast Fourier Transform (FFT) of w 100 (t, x, y) is calculated, which gives a real and an imaginary partŵ r (t, k, l) andŵ i (t, k, l) where k and l are the horizontal wave-numbers along the x-and y-axes. The temporal 1D FFT ofŵ r (t, k, l) andŵ i (t, k, l) is then calculated. It gives two complex fields w r (ω, k, l) andw i (ω, k, l). To obtain the power spectral density, this quantities are recombined as follows: where a is the complex conjugate of a, and R and I are the complex and imaginary parts. Finally an azimuthal average is computed in the (k, l) space to keep only the norm of the spatial wavenumber.

Extraction of the lee waves induced pressure anomaly
This section presents how the pressure anomaly induced by the lee wave phenomena p is extracted from the simulation outputs. The same method has been used to compute horizontal velocity anomalies u and v .
From a vertical along-flow section: Firstly the density is computed from the salinity S and the temperature T with the TEOS-10 equation of state. A spatial high-pass filter is then applied on the time low-passed density. The filter is applied horizontally in the direction of the section at a cutting length λ cut = 10 km. It allows to remove the density background which varies both horizontally and vertically and extract an anomaly of density ρ . Because the filter is applied on the low-pass time-filtered field, and λ cut being chosen to select the along-section small-scale variations of the topography, ρ may be considered as the density anomaly due to lee waves. The CROCO model solving the hydrostatic primitive equations, the pressure anomaly is computed with z=0 is the surface of the ocean where the pressure anomaly is known and equal to zero.
From 3D outputs: A cubic smoothing spline algorithm is applied on the time low-passed density over each vertical level to compute the background density. The fall-off of the smoothing is chosen to keep only the small scales contribution (i.e. O(< 10) km). The anomaly of density is then obtained by removing the background. The pressure anomaly is finally obtained using hydrostatic equation (1).

Finding the typical height H and length scale L of the seamounts
We describe here how we extract typical spatial scales of the bottom seamounts from the bathymetric data, used in the estimation of the dynamical parameters ε and F r lee .
To compute the typical height scale of the bathymetry H along a section (i.e. a 1D bathymetry), we used a peak detection algorithm. Each seamount is defined by a local maximum and two minima (upstream and downstream). The difference between the maximum and the minimum is computed both upstream and downstream. Averaging the two quantities gives a mean height for each seamount. Using the same peak detection, L is defined as the distance between the lows of the bathymetry.
For a 2D bathymetry, the peak detection is performed on the de-trended topography along the x-and ydirections. The procedure follows the same line as in the 1D case: the peak/lows detection is performed along both dimensions, and the resulting 2D maps of H and L from each computation are averaged (dimension-wise) and interpolated on a regular grid.