Arteries carrying warm blood outward and veins carrying cooled blood back run side by side along the limb, exchanging heat directly instead of losing it all to the air. Along the vessel length x (0 = core, 1 = extremity), for artery temperature Ta and vein temperature Tv:
∂T_a/∂t = -v·∂T_a/∂x - k·(T_a - T_v)
∂T_v/∂t = +v·∂T_v/∂x + k·(T_a - T_v) - L(x)·(T_v - T_amb)
L(x) = loss coefficient, largest near the extremity (x→1)
k is the countercurrent coupling — how tightly the two vessels are thermally bonded (the anatomical "rete mirabile" seen in arctic mammal legs, penguin feet, and tuna/whale flippers). v is proportional to blood flow rate. The top panel is a draggable/zoomable schematic of the limb; the bottom panel plots Ta(x) and Tv(x) live along its length.
- Ambient temperature — colder air pulls more heat out at the extremity, cooling the returning vein.
- Core body temperature — sets the fixed temperature blood enters the limb at.
- Blood flow rate — faster flow (v) advects heat down the limb faster, giving the exchanger less time per unit length to trade heat.
- Countercurrent coupling — the exchange strength k; drag to 0% to see the "no adaptation" case where the whole limb runs near core temperature and loses far more heat.
- Countercurrent ON/OFF — toggles whether the vein runs backward (real countercurrent) or forward alongside the artery (co-current, the much less effective arrangement); compare "core heat retained" between the two.
Core heat retained is the fraction of the worst-case heat-loss rate (the whole vein sitting uniformly at core temperature, exposed to the air with no protection at all) that is currently being avoided. It is computed from the actual heat flowing out to the ambient air along the whole vessel, smoothed over a couple of seconds — not from a single point's temperature, because in the co-current arrangement the node nearest the core is the vein's inlet, not its outlet, and sampling it directly (as a naive single-point read would) makes co-current misleadingly look as good as countercurrent. Measuring the actual heat given up to the air fixes that: it correctly shows countercurrent retaining substantially more core heat than the co-current comparison, matching the point of the adaptation — an arctic fox can stand on snow with a foot only a few degrees above freezing while its core stays at 38 °C, because almost all the heat that would otherwise leave with the returning blood is recaptured before it ever reaches the paw.