Why is the Bible seemingly silent on the love which the God the Son has for God the Father? I assume that each member of the Godhead loves each other member of the Godhead. God is, after all, love (1 John 4:8 and 16). 

The Gospels are quite explicit regarding the love which God the Father has for God the Son. Jesus himself makes this very clear (John 3:35; and 5:20). Moreover, the voice coming out of heaven at Jesus' baptism addresses the Son as "My beloved Son" (Luke 3:22; Matthew 3:17; and Mark 1:11). (True enough, the voice from heaven is not linked specifically to the Father, but the word Beloved is clearly linked to the Son, not the Father or the Spirit.)

One could say quite accurately that Jesus proved his love for the Father in everything he did and said while he was on earth. On more than one occasion, Jesus made this devotion to his Father's will quite explicit:


  "My food is to do the will of Him who sent Me and to accomplish His work" (John 4:34). 


And


  ". . . he that sent me is with me: the Father hath not left me alone; for I do always those things that please him" (John 8:29 KJV). 


Logically, then, one could also say that Jesus proved  he loved the Father by doing His will and by obeying Him. In the same way the apostle James emphasized the role which works play in the faith/works duo, perhaps by analogy Jesus in his way of life emphasized the role which loving obedience played in the love/works duo. 

In this regard, the apostle John, the "apostle of love" and "the disciple whom Jesus loved," said, 


  We know love by this, that He laid down His life for us; and we ought to lay down our lives for the brethren. Little children, let us not love with word or with tongue, but in deed and truth" (1 John 3:17-18).


In other words, saying "I love you" to someone is much easier than demonstrating your love for the same person by performing a loving action such as feeding or clothing him, visiting him when he is on his sickbed, comforting him when he has recently lost a loved one through death, or forgiving him when he has sinned against you. 

I have no doubt that Jesus loves his Father, but I am curious as to why that love is not stated as explicitly in the Gospels as is the love of the Father for his Son. I suppose a good answer to my question would draw from both Testaments and would be framed in terms which are consistent with the Father-Son relationship which is foreshadowed in the Tanakh (e.g., in Psalm 2) but revealed more fully in the New Covenant.
I believe that the answer to this is something related to the narrative's approach.
We can infer from what we know about the godhead, that the Son loves the Father, but the focus in the gospels narrative is not that, the focus is that the Father loves the Son, then, the Son loves us and the Son obey the Father. I'll try to examplify this with some text.
First: we know that the Son loves the Father because:

this is love: that we walk in obedience to his commands. (2 John 1:5)

Even Christ's obedience (that is found everywhere, like John 4:34, Philippians 2:8, Hebrews 5:8, Romans 5:9, etc, etc) testify his love for the Father. But, as i said, the scriture approaches this relation upside down, not from the Son to the Father, but from the Father to the Son. We read:

[...] the Father has loved me, so have I loved you. Now remain in my
love. (John 15:9)

As you said, this is the main perspective, and the reason i see is: this is what we need to understand. I think this verse in John 15:9 shows us the emphasis of the Bible's narrative. The Father loves the son, the Son loves us. This thematic is stressed all over, for example:

For the Father loves the Son and shows him all he does. (John 5:20)

and

[...] just as Christ loved us [...] (Ephesians 5:2)


I know this alone isn't a sound argument, so let us check another examples:
Natural marriage
To be quick: husbands are often told to love their wives

Husbands, love your wives and do not be harsh with them.(Colossians 3:19)

and

Husbands, love your wives, just as Christ loved the church and gave himself up for her (Ephesians 5:25)

But wives are always told not to love, but to respect and submit

Wives, submit yourselves to your husbands, as is fitting in the Lord. (Colossians 3:18)

and

Wives, submit yourselves to your own husbands as you do to the Lord (Ephsians 5:22)

So, aren't wives suposed to love their husbands? Yes they are, but i think the Bible is silent on that matter because of the text's approch. I think what husbands need to aknowledge is love and wives is subjection.
Parents and children
The same issue we find into children -&gt; parents realtion. &quot;Children, obey your parents&quot; is the instruction of Colossians 3:20 and Ephasians 6:1. Is not a message of love, is a message of obedience, which is, in fact, the most precise manifestation of love, as we saw in the beginning of this study. So, a son is suposed to love his father? Of course, but what he needs to aknowledge from scripture is not love towards his father, but obedience.
Likewise, Christ, the Son, is shown not loving the Father in a theoretical way, but showing his practical love in the form of obedience. This is what we need to aknowledge, because this is the main focus of God's inpired word, regarding the relation Son-&gt;Father: He obeys.
Old Testment
I was about to sleep when came to my mind a very insightful thought. The first mention of love in the Bible happens when God say to Abraham

Take now your son, your only son Isaac, whom you love (Genesis 21:12)

This is an example of love in this direction: Abraham-&gt;Isaac. But the second mention of love is this

And Isaac [...] took Rebekah, and she became his wife; and he loved her. (Genesis 24:67)

This is an example of love in this direction: Isaac-&gt;Rebekah.
If we take Abraham as a figure of God and Isaac as a figure of the Son and Rebekah as a figure fo the Church, we have this: God loved the Son. The Son loved the Church. You see? This is always the natural flow. The next mention of love in the Bible is when Isaac loves Esau in Genesis 25:28, and so forward.
With maybe some little exceptions, the love always take this flow into all the Scriptures. Father-&gt;Son-&gt;Wife (Thank you, making this research really opened my mind)
Conclusion
At this point, i think we can say that the lack of mentions about the love of the Son towards the Father is really supplied in the plenty of mentions of the Son's obedience. And secondly, i think this &quot;silence&quot; just follows the approach and thematic of the whole Bible argumentation flow about love: The Father loves the Son, the Son obeys the Father and the Son loves his Wife.
@paul-chernoch's answer reminded me of John 14:31, maybe the only mention on the love of the Son towards the Father, and guess what this passage says

I love the Father and do exactly what my Father has commanded me.

Again we see this realtion: i love, therefore, i obey. This is beautiful.