Before we analyze the spatial pattern of bottom pressure changes in the IPCC-A1B scenario, we develop a simple conceptual model that explains to first order how a deep ocean warming can change bottom pressures at depths below and above the steric anomaly (Figure 1). In this approach, we assume that density changes occur uniformly in a certain depth layer, i.e. the density (and thus steric) anomalies are a function of depth only. The total ocean mass does not vary in time. To derive the model, it is sufficient to consider one layer at depth zi with a height hi and areal extent Ai, in an ocean with just three layers. Warming of this layer causes a negative density anomaly ρi′ (Figure 1a), corresponding to a positive specific volume anomaly δi = ρi′/ρ0. Thus, the specific volume anomaly δi would raise the sea surface throughout the horizontal extent of layer i by δihi, and lead to a sharp SSH gradient where the bathymetry becomes shallower (Figure 1b). With no forces present to balance this SSH gradient, we can assume that fast barotropic gravity waves immediately distribute the steric anomaly from layer zi evenly across the entire ocean surface area As (Figure 1c). At this point, the deep warming and concurrent thermal expansion has led to a uniform global sea level rise, but mass redistribution within the basin has led to non-uniform bottom pressure changes. The gain in bottom pressure for the upper shallow layer is δihi (Ai/As), while the loss in bottom pressure for the layer zi and each layer below is δihi[1 − (Ai/As)]. Since mass is conserved, the sum of all bottom pressure changes is zero (e.g., in Figure 1, substitute Ai/As = 2/3, and sum up for all three layers). Generalizing this mechanism to n layers, and allowing for steric anomalies in all layers, we derive a discrete formulation for horizontally uniform, but vertically varying bottom pressure changes:
where i = 1, …, n counts downward from the surface, and ηs′(i) = (Ai/As) δihi represents the steric sea level change contribution from layer i. Equation (3) states that a layer gains mass from the expansion of all layers below, and loses mass from its own expansion and expansion of all layers above. Equivalently, this statement also holds for negative expansion (contraction, or cooling), exchanging gains with losses and vice versa. Note, however, that steric expansion through warming is a very slow process compared to barotropic adjustment timescales in the real ocean. Therefore, the redistribution would always be immediate, and a SSH gradient as described in Figure 1b would not build up.
 In order to estimate the magnitude of bottom pressure anomalies from ocean warming as a function of depth and time in an ocean with realistic topography, we use equation (3) and apply it to the horizontally averaged steric changes in our IPCC-A1B scenario for each model layer (Figure 2). As the warming penetrates deeper into the ocean over time, positive bottom pressure anomalies develop above the warming, with highest amplitudes at the shallowest depths. The positive anomaly around 2500 m depth for the years 2001 to 2020 is due to a relative cooling (and thus negative steric anomaly) between 200 m and 1500 m depth, and might be linked to aerosol induced cooling carried over from the 20th century. After the year 2020, positive bottom pressure anomalies do not reach deeper than 2200 m, which approximately corresponds to the maximum depth where steric anomalies occur (with the exception of the Southern Ocean).