Adjusted Sea Level: We used to call this Gridded Sea Level but changed the name in Sep 2021. Adjusted Sea level (like GSL) is sea level minus two rapidly-varying sea level signals mostly due to barotopic dynamics: astronomical tides and the ocean's response to atmospheric pressure (which is to rise about 10 cm for each 10 hPa fall of pressure). Adjusted Sea Level is thus a measure of the slow modes of the ocean. The slow modes are largely in geostrophic balance, so we can estimate the near-surface velocity from the gradients of ASL. ASL is also a measure of depth-integrated density (and thus ocean heat content), as shown on our Argo pages where ASL is compared with in-situ determinations of steric height anomaly.
Terminology: The once-traditional term 'adjusted' is commonly omitted by many agencies and oceanographers. We did this, too, then regretted it, because dropping the 'adjusted' leaves no short name for 'unadjusted sea level', which is the quantity most relevant to users interested in coastal inundation (see our Australia/NZ maps of Non-tidal Sea Level). Another candidate name for Adjusted Sea Level is Ocean dynamic sea level. Related names include: dynamic height, dynamic topography, and subsurface pressure.
Adjusted Sea Level Anomaly: By 'anomaly', we mean the departure from the long-term (1993-2012) mean. We (like most users of altimetry) subtract the long-term mean of ASL from the altimeter observations in order to remove the ~100m sea level highs and lows (with respect to a smooth surface) mostly due to gravity perturbations associated with sea floor topography. This is done using a correction supplied by the space agencies, which is the Mean Sea Surface (MSS, evaluated along the precise track of the altimeter). Unfortunately this also removes the ~1m scale highs and lows of the Mean Dynamic Topography (MDT) due to the long-term mean of the ocean circulation (that causes the Coral Sea off Queensland to be about 1m higher than waters off Tasmania with respect to the geoid). We add the Bluelink ocean model estimate of the MDT to our altimetry-derived maps of ASLA (formerly known as GSLA) to produce ASL, from which we estimate velocities that include the mean. Most users, however, are more interested in the anomaly of sea level (adjusted or not) than the more abstract concept of sea level with respect to the geoid. Showing the anomaly also allows use of a more restricted colour bar, and to show the along-track altimetry data closer to its supplied form.