Why is there an electric field around this closed electric circuit? I have difficulty understanding the picture below. I can understand the green magnetic fields. But why is there an outside, red, electric field? Isn't the electric field guided entirely through the wire? Are there arguments or measurements showing that there is an electric field as depicted? The blue vector is the Poynting vector, associated with energy flux.
I can understand an electric field around the battery if no current runs. But doesn't this disappear when the current runs?

 A: The wires have a bit of self capacitance. This self capacitance means that when they are at a given voltage then they gain a corresponding amount of net charge. Having that net charge leads to an E field in the space around the wires, as drawn.
A: 
I can understand an electric field around the battery if no current
runs. But doesn't this disappear when the current runs?

No.
The difference is the electric field in the conductor exerts a force on the free electrons  creating current, whereas outside the conductor current cannot flow because of the lack of mobile charges. But the electric field still exists.
Hope this helps.
A: $$\mathbf E = -\frac{dV}{dx}$$ Inside (because there is an internal resistance) and around the battery, it points from the highest (+) to the lowest potential (-) as showed in the picture. In the resistor, for the same definition, it has the same direction of the current.
Imagine a small positive test charge outside and close to the resistor. It is repelled by the upper side and attracted by the lower side. The same type of field configuration is also present inside the components, but is not restricted only to them.
A: 
Isn't the electric field guided entirely through the wire?

The wires are (very nearly) perfect conductors, and therefore the electric field within the wires is (very nearly) zero.

why is there an outside, red, electric field?

Because there is no field inside the wires, the wires are (very nearly) equipotential volumes. And you know there is a potential difference between the upper wire and the lower wire, of (let's say) 1.5 V, because one is connected to the anode of the battery and the other is connected to the cathode.
Therefore you know that if you take a point a on the upper wire and a point b on the lower wire and calculate (based on the definition of electrostatic potential difference)
$$V_{ab} = -\int_b^a\mathbf{E}\cdot d\mathbf{\ell}$$
you will get 1.5 V.
That tells you that the electric field in the region between the wires must be non-zero, and must point from the upper wire to the lower wire.

I can understand an electric field around the battery if no current runs. But doesn't this disappear when the current runs?

No. The electric field is there if there is a potential difference between the wires. It doesn't depend on whether current is running or not. (The magnetic fields, of course, do depend on the current)
